A broken circuit model for chromatic homology theories

}, volume={104}, ISSN={["1095-9971"]}, url={http://dx.doi.org/10.1016/j.ejc.2022.103538}, DOI={10.1016/j.ejc.2022.103538}, abstractNote={Using the tools of algebraic Morse theory, and the thin poset approach to constructing homology theories, we give the categorification of Whitney’s broken circuit theorem for the chromatic polynomial, and for Stanley’s chromatic symmetric function.}, journal={EUROPEAN JOURNAL OF COMBINATORICS}, author={Chandler, Alex and Sazdanovic, Radmila}, year={2022}, month={Aug} } @article{levitt_hajij_sazdanovic_2022, title={Big data approaches to knot theory: Understanding the structure of the Jones polynomial}, volume={11}, ISSN={["1793-6527"]}, url={http://dx.doi.org/10.1142/s021821652250095x}, DOI={10.1142/S021821652250095X}, abstractNote={In this paper, we examine the properties of the Jones polynomial using dimensionality reduction learning techniques combined with ideas from topological data analysis. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on the Jones polynomial data, we find that it can be viewed as an approximately three-dimensional subspace, that this description is surprisingly stable with respect to the filtration by the crossing number, and that the results suggest further structures to be examined and understood.}, journal={JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS}, author={Levitt, Jesse S. F. and Hajij, Mustafa and Sazdanovic, Radmila}, year={2022}, month={Nov} } @article{sazdanovic_scofield_2022, title={Extremal Khovanov homology and the girth of a knot}, volume={10}, ISSN={["1793-6527"]}, url={http://dx.doi.org/10.1142/s0218216522500833}, DOI={10.1142/S0218216522500833}, abstractNote={We show that Khovanov link homology is trivial in a range of gradings and utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and the chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper, we define the girth of a link, discuss relations to other knot invariants, and describe possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that the girth of the corresponding all-A graph is equal to [Formula: see text], then the girth of the link is equal to [Formula: see text]}, journal={JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS}, author={Sazdanovic, Radmila and Scofield, Daniel}, year={2022}, month={Oct} } @inbook{gasparovic_gommel_purvine_sazdanovic_wang_wang_ziegelmeier_2022, title={Local Versus Global Distances for Zigzag and Multi-Parameter Persistence Modules}, url={http://dx.doi.org/10.1007/978-3-030-95519-9_3}, DOI={10.1007/978-3-030-95519-9_3}, booktitle={Association for Women in Mathematics Series}, author={Gasparovic, Ellen and Gommel, Maria and Purvine, Emilie and Sazdanovic, Radmila and Wang, Bei and Wang, Yusu and Ziegelmeier, Lori}, year={2022} } @article{aslam_ardanza-trevijano_xiong_arsuaga_sazdanovic_2022, title={TAaCGH Suite for Detecting Cancer-Specific Copy Number Changes Using Topological Signatures}, volume={24}, ISSN={["1099-4300"]}, url={https://www.mdpi.com/1099-4300/24/7/896}, DOI={10.3390/e24070896}, abstractNote={Copy number changes play an important role in the development of cancer and are commonly associated with changes in gene expression. Persistence curves, such as Betti curves, have been used to detect copy number changes; however, it is known these curves are unstable with respect to small perturbations in the data. We address the stability of lifespan and Betti curves by providing bounds on the distance between persistence curves of Vietoris–Rips filtrations built on data and slightly perturbed data in terms of the bottleneck distance. Next, we perform simulations to compare the predictive ability of Betti curves, lifespan curves (conditionally stable) and stable persistent landscapes to detect copy number aberrations. We use these methods to identify significant chromosome regions associated with the four major molecular subtypes of breast cancer: Luminal A, Luminal B, Basal and HER2 positive. Identified segments are then used as predictor variables to build machine learning models which classify patients as one of the four subtypes. We find that no single persistence curve outperforms the others and instead suggest a complementary approach using a suite of persistence curves. In this study, we identified new cytobands associated with three of the subtypes: 1q21.1-q25.2, 2p23.2-p16.3, 23q26.2-q28 with the Basal subtype, 8p22-p11.1 with Luminal B and 2q12.1-q21.1 and 5p14.3-p12 with Luminal A. These segments are validated by the TCGA BRCA cohort dataset except for those found for Luminal A.}, number={7}, journal={ENTROPY}, author={Aslam, Jai and Ardanza-Trevijano, Sergio and Xiong, Jingwei and Arsuaga, Javier and Sazdanovic, Radmila}, year={2022}, month={Jul} } @article{khovanov_sazdanovic_2021, title={Diagrammatic categorification of the Chebyshev polynomials of the second kind}, volume={225}, ISSN={["1873-1376"]}, DOI={10.1016/j.jpaa.2020.106592}, abstractNote={We develop a diagrammatic categorification of the polynomial ring Z[x], based on a geometrically defined graded algebra. This construction generalizes to categorification of some special functions, such as Chebyshev polynomials. Diagrammatic algebras featured in these categorifications lead to the first topological interpretations of the Bernstein-Gelfand-Gelfand reciprocity property.}, number={6}, journal={Journal of Pure and Applied Algebra}, author={Khovanov, M. and Sazdanovic, R.}, year={2021}, month={Jun}, pages={1006592} } @article{khovanov_sazdanovic_2021, title={Diagrammatic categorification of the Chebyshev polynomials of the second kind}, volume={225}, url={http://www.sciencedirect.com/science/article/pii/S0022404920302930}, DOI={https://doi.org/10.1016/j.jpaa.2020.106592}, abstractNote={We develop a diagrammatic categorification of the polynomial ring Z[x], based on a geometrically defined graded algebra. This construction generalizes to categorification of some special functions, such as Chebyshev polynomials. Diagrammatic algebras featured in these categorifications lead to the first topological interpretations of the Bernstein-Gelfand-Gelfand reciprocity property.}, number={6}, journal={Journal of Pure and Applied Algebra}, author={Khovanov, Mikhail and Sazdanovic, Radmila}, year={2021}, pages={106592} } @inbook{adams_flapan_henrich_kauffman_ludwig_nelson_2021, title={Encyclopedia of Knot Theory}, url={http://dx.doi.org/10.1201/9781138298217}, DOI={10.1201/9781138298217}, abstractNote={theory is a fascinating mathematical subject, with multiple links to theoretical physics. This enyclopedia is filled with valuable information on a rich and fascinating subject. – Ed Witten, Recipient of the Fields Medal I spent a pleasant afternoon perusing the Encyclopedia of Knot Theory. It’s a comprehensive compilation of clear introductions to both classical and very modern developments in the field. It will be a terrific resource for the accomplished researcher, and will also be an excellent way to lure students, both graduate and undergraduate, into the field. – Abigail Thompson, Distinguished Professor of Mathematics at University of California, Davis Knot theory has proven to be a fascinating area of mathematical research, dating back about 150 years. Encyclopedia of Knot Theory provides short, interconnected articles on a variety of active areas in knot theory, and includes beautiful pictures, deep mathematical connections, and critical applications. Many of the articles in this book are accessible to undergraduates who are working on research or taking an advanced undergraduate course in knot theory. More advanced articles will be useful to graduate students working on a related thesis topic, to researchers in another area of topology who are interested in current results in knot theory, and to scientists who study the topology and geometry of biopolymers. Features Provides material that is useful and accessible to undergraduates, postgraduates, and full-time researchers Topics discussed provide an excellent catalyst for students to explore meaningful research and gain confidence and commitment to pursuing advanced degrees Edited and contributed by top researchers in the field of knot theory}, booktitle={Chapman and Hall/CRC}, author={Adams, Colin and Flapan, Erica and Henrich, Allison and Kauffman, Louis H. and Ludwig, Lewis D. and Nelson, Sam}, year={2021}, month={Feb} } @article{baldwin_dowlin_levine_lidman_sazdanovic_2021, title={Khovanov homology detects the figure‐eight knot}, volume={53}, ISSN={0024-6093 1469-2120}, url={http://dx.doi.org/10.1112/blms.12467}, DOI={10.1112/blms.12467}, abstractNote={We use Dowlin's spectral sequence from Khovanov homology to knot Floer homology to prove that reduced Khovanov homology with rational coefficients detects the figure-eight knot.}, number={3}, journal={Bulletin of the London Mathematical Society}, publisher={Wiley}, author={Baldwin, John A. and Dowlin, Nathan and Levine, Adam Simon and Lidman, Tye and Sazdanovic, Radmila}, year={2021}, month={Jan}, pages={871–876} } @inbook{russell_sazdanovic_2021, title={Mathematics and Art: Unifying Perspectives}, ISBN={978-3-319-57071-6}, DOI={https://doi.org/10.1007/978-3-319-57072-3}, booktitle={Handbook of the Mathematics of the Arts and Sciences}, publisher={Springer}, author={Russell, H. and Sazdanovic, R.}, editor={Sriraman, B.Editor}, year={2021}, pages={497–525} } @inbook{caprau_gonzález_lee_lowrance_sazdanović_zhang_2021, title={On Khovanov Homology and Related Invariants}, url={http://dx.doi.org/10.1007/978-3-030-80979-9_6}, DOI={10.1007/978-3-030-80979-9_6}, abstractNote={This paper begins with a survey of some applications of Khovanov homology to low-dimensional topology, with an eye toward extending these results to \(\mathfrak {sl}(n)\) homologies. We extend Levine-Zemke’s ribbon concordance obstruction from Khovanov homology to \(\mathfrak {sl}(n)\) foam homologies for n ≥ 2, including the universal \(\mathfrak {sl}(2)\) and \(\mathfrak {sl}(3)\) foam homology theories. Inspired by Alishahi and Dowlin’s bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral sequences also provide lower bounds on Turaev genus.}, booktitle={Association for Women in Mathematics Series}, author={Caprau, Carmen and González, Nicolle and Lee, Christine Ruey Shan and Lowrance, Adam M. and Sazdanović, Radmila and Zhang, Melissa}, year={2021} } @article{dabkowski_harizanov_kauffman_przytycki_sazdanovic_sikora_2021, title={Preface}, volume={30}, ISSN={["1793-6527"]}, DOI={10.1142/S0218216521020016}, abstractNote={Journal of Knot Theory and Its RamificationsVol. 30, No. 14, 2102001 (2021) No AccessPrefaceMieczyslaw K. Dabkowski, Valentina Harizanov, Louis H. Kauffman, Jozef H. Przytycki, Radmila Sazdanovic, and Adam SikoraMieczyslaw K. DabkowskiUniversity of Texas at Dallas, USA, Valentina HarizanovGeorge Washington University, USA, Louis H. KauffmanUniversity of Illinois at Chicago, USA, Jozef H. PrzytyckiGeorge Washington University, USA, Radmila SazdanovicNorth Carolina State University, USA, and Adam SikoraSUNY Buffalo, USAhttps://doi.org/10.1142/S0218216521020016Cited by:0 Next This article is part of the issue: Special Issue: Dedicated to the 60th Birthday of J. H. PrzvtyckiGuest Editors: M. K. Dabkowski, V. Harizanov, L. H. Kauffman, J. H. Przytycki, R. Sazdanovic and A. Sikora AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetails Recommended Vol. 30, No. 14 Metrics History PDF download}, number={14}, journal={JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS}, author={Dabkowski, Mieczyslaw K. and Harizanov, Valentina and Kauffman, Louis H. and Przytycki, Jozef H. and Sazdanovic, Radmila and Sikora, Adam}, year={2021}, month={Dec} } @article{sazdanovic_summers_2021, title={Torsion in the magnitude homology of graphs}, volume={16}, ISSN={["1512-2891"]}, url={http://dx.doi.org/10.1007/s40062-021-00281-9}, DOI={10.1007/s40062-021-00281-9}, abstractNote={Magnitude homology is a bigraded homology theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced by Leinster. We analyze the structure and implications of torsion in magnitude homology. We show that any finitely generated abelian group may appear as a subgroup of the magnitude homology of a graph, and, in particular, that torsion of a given prime order can appear in the magnitude homology of a graph and that there are infinitely many such graphs. Finally, we provide complete computations of magnitude homology of a class of outerplanar graphs and focus on the ranks of the groups along the main diagonal of magnitude homology.}, number={2}, journal={JOURNAL OF HOMOTOPY AND RELATED STRUCTURES}, author={Sazdanovic, Radmila and Summers, Victor}, year={2021}, month={Jun}, pages={275–296} } @article{chandler_lowrance_sazdanović_summers_2021, title={Torsion in thin regions of Khovanov homology}, volume={74}, ISSN={0008-414X 1496-4279}, url={http://dx.doi.org/10.4153/s0008414x21000043}, DOI={10.4153/S0008414X21000043}, abstractNote={Abstract In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is, links whose Khovanov homology is supported on two adjacent diagonals, are known to contain only $\mathbb {Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported on two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only $\mathbb {Z}_2$ torsion over that range of homological gradings. These conditions are then shown to be met by an infinite family of three-braids, strictly containing all three-strand torus links, thus giving a partial answer to Sazdanović and Przytycki’s conjecture that three-braids have only $\mathbb {Z}_2$ torsion in Khovanov homology. We use these computations and our main theorem to obtain the integral Khovanov homology for all links in this family.}, number={3}, journal={Canadian Journal of Mathematics}, publisher={Canadian Mathematical Society}, author={Chandler, Alex and Lowrance, Adam M. and Sazdanović, Radmila and Summers, Victor}, year={2021}, month={Jan}, pages={1–27} } @inbook{sazdanovic_2021, place={Münster, Germany}, series={Kultur: Forschung und Wissenschaft}, title={Visualizations and visual thinking in mathematics}, ISBN={9783643905352}, booktitle={On visualization: A multicentric critique beyond inforgraphics}, publisher={LIT Verlag}, author={Sazdanovic, R.}, editor={Fiorentini, Erna and Elinks, JamesEditors}, year={2021}, collection={Kultur: Forschung und Wissenschaft} } @book{khovanov_sazdanovic_2020, title={Bilinear pairings on two-dimensional cobordisms and generalizations of the Deligne category}, number={2007.11640}, author={Khovanov, M. and Sazdanovic, R.}, year={2020}, month={Jul} } @book{sazdanovic_scofield_2020, title={Extremal Khovanov homology and the girth of a knot}, number={2003.05074}, author={Sazdanovic, R. and Scofield, D.}, year={2020}, month={Mar} } @inbook{sazdanovic_2020, title={Khovanov Link Homology}, ISBN={9781138297845}, booktitle={Encyclopedia of Knot Theory}, publisher={Chapman & Hall}, author={Sazdanovic, R.}, editor={Adams, C. and Flapan, E. and Henrich, A. and Kauffman, L.H. and Ludwig, L.D. and Nelson, S.Editors}, year={2020}, month={Dec}, pages={Chapter 70} } @inproceedings{caprau_gonzalez_lee_lowrance_sazdanovic_zhang_2020, title={On Khovanov Homology and Related Invariants}, booktitle={Proceedings of the Research Collaboration Conference of the Women in Symplectic and Contact Geometry and Topology}, publisher={Springer}, author={Caprau, C. and Gonzalez, N. and Lee, Christine Ruey Shan and Lowrance, A. and Sazdanovic, R. and Zhang, M.}, year={2020}, month={Feb} } @article{adamaszek_adams_gasparovic_gommel_purvine_sazdanovic_wang_wang_ziegelmeier_2020, title={On homotopy types of Vietoris–Rips complexes of metric gluings}, volume={4}, ISSN={2367-1726 2367-1734}, url={http://dx.doi.org/10.1007/s41468-020-00054-y}, DOI={10.1007/s41468-020-00054-y}, abstractNote={Given a sample of points $X$ in a metric space $M$ and a scale $r>0$, the Vietoris-Rips simplicial complex $\mathrm{VR}(X;r)$ is a standard construction to attempt to recover $M$ from $X$ up to homotopy type. A deficiency of this approach is that $\mathrm{VR}(X;r)$ is not metrizable if it is not locally finite, and thus does not recover metric information about $M$. We attempt to remedy this shortcoming by defining a metric space thickening of $X$, which we call the \emph{Vietoris-Rips thickening} $\mathrm{VR}^m(X;r)$, via the theory of optimal transport. When $M$ is a complete Riemannian manifold, or alternatively a compact Hadamard space, we show that the the Vietoris-Rips thickening satisfies Hausmann's theorem ($\mathrm{VR}^m(M;r)\simeq M$ for $r$ sufficiently small) with a simpler proof: homotopy equivalence $\mathrm{VR}^m(M;r)\to M$ is canonically defined as a center of mass map, and its homotopy inverse is the (now continuous) inclusion map $M\hookrightarrow\mathrm{VR}^m(M;r)$. Furthermore, we describe the homotopy type of the Vietoris-Rips thickening of the $n$-sphere at the first positive scale parameter $r$ where the homotopy type changes.}, number={3}, journal={Journal of Applied and Computational Topology}, publisher={Springer Science and Business Media LLC}, author={Adamaszek, Michał and Adams, Henry and Gasparovic, Ellen and Gommel, Maria and Purvine, Emilie and Sazdanovic, Radmila and Wang, Bei and Wang, Yusu and Ziegelmeier, Lori}, year={2020}, month={May}, pages={425–454} } @inbook{gonzalez_ushakova_sazdanovic_arsuaga_2020, title={Prediction in Cancer Genomics Using Topological Signatures and Machine Learning}, ISBN={9783030434076 9783030434083}, ISSN={2193-2808 2197-8549}, url={http://dx.doi.org/10.1007/978-3-030-43408-3_10}, DOI={10.1007/978-3-030-43408-3_10}, abstractNote={Copy Number Aberrations, gains and losses of genomic regions, are a hallmark of cancer and can be experimentally detected using microarray comparative genomic hybridization (aCGH). In previous works, we developed a topology based method to analyze aCGH data whose output are regions of the genome where copy number is altered in patients with a predetermined cancer phenotype. We call this method Topological Analysis of array CGH (TAaCGH). Here we combine TAaCGH with machine learning techniques to build classifiers using copy number aberrations. We chose logistic regression on two different binary phenotypes related to breast cancer to illustrate this approach. The first case consists of patients with over-expression of the ERBB2 gene. Over-expression of ERBB2 is commonly regulated by a copy number gain in chromosome arm 17q. TAaCGH found the region 17q11-q22 associated with the phenotype and using logistic regression we reduced this region to 17q12-q21.31 correctly classifying 78% of the ERBB2 positive individuals (sensitivity) in a validation data set. We also analyzed over-expression in Estrogen Receptor (ER), a second phenotype commonly observed in breast cancer patients and found that the region 5p14.3-12 together with six full arms were associated with the phenotype. Our method identified 4p, 6p and 16q as the strongest predictors correctly classifying 76% of ER positives in our validation data set. However, for this set there was a significant increase in the false positive rate (specificity). We suggest that topological and machine learning methods can be combined for prediction of phenotypes using genetic data.}, booktitle={Topological Data Analysis}, publisher={Springer International Publishing}, author={Gonzalez, Georgina and Ushakova, Arina and Sazdanovic, Radmila and Arsuaga, Javier}, year={2020}, pages={247–276} } @book{chandler_sazdanovic_2019, title={A Broken Circuit Model for Chromatic Homology Theories}, number={1911.13226}, author={Chandler, A. and Sazdanovic, R.}, year={2019} } @article{adams_gordon_jones_kauffman_lambropoulou_millett_przytycki_ricca_sazdanovic_2019, title={Knots, low-dimensional topology and applications Preface}, volume={28}, ISSN={["1793-6527"]}, DOI={10.1142/S0218216519020012}, abstractNote={Journal of Knot Theory and Its RamificationsVol. 28, No. 11, 1902001 (2019) No AccessPreface: Knots, low-dimensional topology and applicationshttps://doi.org/10.1142/S0218216519020012Cited by:0 Next This article is part of the issue: Special Issue: Knots, Low-Dimensional Topology and Applications — Part I AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetails Recommended Vol. 28, No. 11 Metrics History PDF download}, number={11}, journal={JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS}, author={Adams, Colin C. and Gordon, Cameron McA and Jones, Vaughan F. R. and Kauffman, Louis H. and Lambropoulou, Sofia and Millett, Kenneth and Przytycki, Jozef H. and Ricca, Renzo and Sazdanovic, Radmila}, year={2019}, month={Oct} } @article{adams_gordon_jones_kauffman_lambropoulou_millett_przytycki_ricca_sazdanovic_2019, title={Knots, low-dimensional topology and applications Preface}, volume={28}, ISSN={["1793-6527"]}, DOI={10.1142/S0218216519030019}, number={13}, journal={JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS}, author={Adams, Colin C. and Gordon, Cameron McA and Jones, Vaughan F. R. and Kauffman, Louis H. and Lambropoulou, Sofia and Millett, Kenneth and Przytycki, Jozef H. and Ricca, Renzo and Sazdanovic, Radmila}, year={2019}, month={Nov} } @book{gasparovic_gommel_purvine_sazdanovic_wang_wang_ziegelmeier_2019, title={Local Versus Global Distances for Zigzag Persistence Modules}, number={1903.08298}, author={Gasparovic, E. and Gommel, M. and Purvine, E. and Sazdanovic, R. and Wang, B. and Wang, Y. and Ziegelmeier, L.}, year={2019} } @article{cooper_silva_sazdanovic_2019, title={On configuration spaces and simplicial complexes}, volume={25}, journal={New York J. Math}, author={Cooper, Andrew A and Silva, Vin and Sazdanovic, Radmila}, year={2019}, pages={723–744} } @book{chandler_sazdanovic_stella_yip_2019, title={On the Strength of Chromatic Symmetric Homology for graphs}, number={1911.13297}, author={Chandler, A. and Sazdanovic, R. and Stella, S. and Yip, M.}, year={2019} } @article{nelson_oyamaguchi_sazdanovic_2019, title={Psyquandles, Singular Knots and Pseudoknots}, volume={42}, ISSN={["0387-3870"]}, DOI={10.3836/tjm/1502179287}, abstractNote={We generalize the notion of biquandles to psyquandles and use these to define invariants of oriented singular links and pseudolinks. In addition to psyquandle counting invariants, we introduce Alexander psyquandles and corresponding invariants such as Alexander psyquandle polynomials and Alexander-Gröbner psyquandle invariants of oriented singular knots and links. We consider the relationship between Alexander psyquandle colorings of pseudolinks and $p$-colorings of pseudolinks. As a special case we define a generalization of the Alexander polynomial for oriented singular links and pseudolinks we call the Jablan polynomial and compute the invariant for all pseudoknots with up to five crossings and all 2-bouquet graphs with up to 6 classical crossings.}, number={2}, journal={TOKYO JOURNAL OF MATHEMATICS}, author={Nelson, Sam and Oyamaguchi, Natsumi and Sazdanovic, Radmila}, year={2019}, month={Dec}, pages={405–429} } @article{gasparovic_gommel_purvine_sazdanovic_wang_wang_ziegelmeier_2019, title={The relationship between the intrinsic Čech and persistence distortion distances for metric graphs}, url={https://jocg.org/index.php/jocg/article/view/3082}, DOI={10.20382/JOCG.V10I1A16}, abstractNote={Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric graphs in the context of comparison, we are interested in determining the relative discriminative capabilities of two topology-based distances between a pair of arbitrary finite metric graphs: the persistence distortion distance and the intrinsic Cech distance. We explicitly show how to compute the intrinsic Cech distance between two metric graphs based solely on knowledge of the shortest systems of loops for the graphs. Our main theorem establishes an inequality between the intrinsic Cech and persistence distortion distances in the case when one of the graphs is a bouquet graph and the other is arbitrary. The relationship also holds when both graphs are constructed via wedge sums of cycles and edges.}, journal={Journal of Computational Geometry}, publisher={Journal of Computational Geometry}, author={Gasparovic, Ellen and Gommel, Maria and Purvine, Emilie and Sazdanovic, Radmila and Wang, Bei and Wang, Yusu and Ziegelmeier, Lori}, year={2019}, month={Nov}, pages={Vol. 10 No. 1 (2019)} } @article{gasparovic_gommel_purvine_sazdanovic_wang_wang_ziegelmeier_2019, title={The relationship between the intrinsic Čech and persistence distortion distances for metric graphs}, volume={10}, number={1}, journal={Journal of Computational Geometry}, author={Gasparovic, E. and Gommel, M. and Purvine, E. and Sazdanovic, R. and Wang, B. and Wang, Y. and Ziegelmeier, L.}, year={2019}, pages={477–499} } @book{sazdanovic_summers_2019, title={Torsion in the Magnitude homology of graphs}, number={1912.13483}, author={Sazdanovic, R. and Summers, V.}, year={2019} } @article{gasparovic_gommel_purvine_sazdanovic_wang_wang_ziegelmeier_2018, title={A Complete Characterization of the One-Dimensional Intrinsic Čech Persistence Diagrams for Metric Graphs}, DOI={10.1007/978-3-319-89593-2_3}, abstractNote={Metric graphs are special types of metric spaces used to model and represent simple, ubiquitous, geometric relations in data such as biological networks, social networks, and road networks. We are interested in giving a qualitative description of metric graphs using topological summaries. In particular, we provide a complete characterization of the one-dimensional intrinsic Čech persistence diagrams for finite metric graphs using persistent homology. Together with complementary results by Adamaszek et al., which imply the results on intrinsic Čech persistence diagrams in all dimensions for a single cycle, our results constitute the important steps toward characterizing intrinsic Čech persistence diagrams for arbitrary finite metric graphs across all dimensions.}, journal={Research in Computational Topology}, publisher={Springer International Publishing}, author={Gasparovic, Ellen and Gommel, Maria and Purvine, Emilie and Sazdanovic, Radmila and Wang, Bei and Wang, Yusu and Ziegelmeier, Lori}, year={2018}, pages={33–56} } @article{sazdanovic_yip_2018, title={A categorification of the chromatic symmetric function}, volume={154}, ISSN={0097-3165}, url={http://dx.doi.org/10.1016/j.jcta.2017.08.014}, DOI={10.1016/j.jcta.2017.08.014}, abstractNote={The Stanley chromatic symmetric function XG of a graph G is a symmetric function generalization of the chromatic polynomial and has interesting combinatorial properties. We apply the ideas from Khovanov homology to construct a homology theory of graded Sn-modules, whose graded Frobenius series FrobG(q,t) specializes to the chromatic symmetric function at q=t=1. This homology theory can be thought of as a categorification of the chromatic symmetric function, and it satisfies homological analogues of several familiar properties of XG. In particular, the decomposition formula for XG discovered recently by Orellana, Scott, and independently by Guay-Paquet, is lifted to a long exact sequence in homology.}, journal={Journal of Combinatorial Theory, Series A}, publisher={Elsevier BV}, author={Sazdanovic, Radmila and Yip, Martha}, year={2018}, month={Feb}, pages={218–246} } @article{sazdanovic_2018, title={Khovanov homology and torsion}, ISBN={["978-3-11-057148-6"]}, DOI={10.1515/9783110571493-005}, abstractNote={The present chapter gives an overview on results for discrete knot energies. These discrete energies are designed to make swift numerical computations and thus open the field to computational methods. Additionally, they provide an independent, geometrically pleasing and consistent discrete model that behaves similarly to the original model. We will focus on Mobius energy, integral Menger curvature and thickness.}, journal={NEW DIRECTIONS IN GEOMETRIC AND APPLIED KNOT THEORY}, author={Sazdanovic, Radmila}, year={2018}, pages={125–137} } @article{sazdanovic_scofield_2018, title={Patterns in Khovanov link and chromatic graph homology}, volume={27}, ISSN={["1793-6527"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85044463913&partnerID=MN8TOARS}, DOI={10.1142/s0218216518400072}, abstractNote={Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. We discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme–Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.}, number={3}, journal={JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS}, publisher={World Scientific Pub Co Pte Lt}, author={Sazdanovic, Radmila and Scofield, Daniel}, year={2018}, month={Mar} } @article{dabkowski_harizanov_kauffman_przytycki_sazdanovic_sikora_2018, title={Special Issue - Devoted to the 60th birthday of J. H. Przytycki; Guest Editors: M. K. Dabkowski, V. Harizanov, L. H. Kauffman, J. H. Przytycki, R. Sazdanovic and A. Sikora Preface}, volume={27}, number={3}, journal={Journal of Knot Theory and its Ramifications}, author={Dabkowski, M. K. and Harizanov, V. and Kauffman, L. H. and Przytycki, J. H. and Sazdanovic, R. and Sikora, A.}, year={2018} } @inproceedings{adamaszek_adams_gasparovic_gommel_purvine_sazdanovic_wang_wang_ziegelmeier_2018, title={Vietoris-rips and čech complexes of metric gluings}, volume={99}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85048959293&partnerID=MN8TOARS}, DOI={10.4230/LIPIcs.SoCG.2018.3}, abstractNote={We study Vietoris-Rips and Cech complexes of metric wedge sums and metric gluings. We show that the Vietoris-Rips (resp. Cech) complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris-Rips (resp. Cech) complexes. We also provide generalizations for certain metric gluings, i.e. when two metric spaces are glued together along a common isometric subset. As our main example, we deduce the homotopy type of the Vietoris-Rips complex of two metric graphs glued together along a sufficiently short path. As a result, we can describe the persistent homology, in all homological dimensions, of the Vietoris-Rips complexes of a wide class of metric graphs.}, booktitle={Leibniz International Proceedings in Informatics, LIPIcs}, author={Adamaszek, M. and Adams, H. and Gasparovic, E. and Gommel, M. and Purvine, E. and Sazdanovic, R. and Wang, B. and Wang, Y. and Ziegelmeier, L.}, year={2018}, pages={32–315} } @inproceedings{adamaszek_adams_gasparovic_gommel_purvine_sazdanovic_wang_wang_ziegelmeier_2018, place={Wadern, Germany}, series={LIPIcs - Leibniz International Proceedings in Informatics}, title={Vietoris–Rips and Cech Complexes of Metric Gluings}, volume={3}, number={1-3}, booktitle={34th International Symposium on Computational Geometry (SoCG 2018)}, publisher={Wadern Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH}, author={Adamaszek, M. and Adams, H. and Gasparovic, E. and Gommel, M. and Purvine, E. and Sazdanovic, R. and Wang, B. and Wang, Y. and Ziegelmeier, L.}, editor={Speckmann, Bettina and Toth, Csaba D.Editors}, year={2018}, pages={218–246}, collection={LIPIcs - Leibniz International Proceedings in Informatics} } @article{lowrance_sazdanovic_2017, title={Chromatic homology, Khovanov homology, and torsion}, volume={222}, ISSN={["1879-3207"]}, DOI={10.1016/j.topol.2017.02.078}, abstractNote={In the first few homological gradings, there is an isomorphism between the Khovanov homology of a link and the categorification of the chromatic polynomial of a graph related to the link. In this article, we show that all torsion in the categorification of the chromatic polynomial is of order two, and hence all torsion in Khovanov homology in the gradings where the isomorphism is defined is of order two. We also prove that odd Khovanov homology is torsion-free in its first few homological gradings.}, journal={TOPOLOGY AND ITS APPLICATIONS}, publisher={Elsevier BV}, author={Lowrance, Adam M. and Sazdanovic, Radmila}, year={2017}, month={May}, pages={77–99} } @inproceedings{guan_tang_krim_keiser_rindos_sazdanovic_2016, title={A topological collapse for document summarization}, volume={2016-August}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84984653594&partnerID=MN8TOARS}, DOI={10.1109/spawc.2016.7536867}, abstractNote={As a useful tool to summarize documents, keyphrase extraction extracts a set of single or multiple words, called keyphrases, that capture the primary topics discussed in a document. In this paper we propose DoCollapse, a topological collapse-based unsupervised keyphrase extraction method that relies on networking document by semantic relatedness of candidate keyphrases. A semantic graph is built with candidates keyphrases as vertices and then reduced to its core using topological collapse algorithm to facilitate final keyphrase selection. Iteratively collapsing dominated vertices aids in removing noisy candidates and revealing important points. We conducted experiments on two standard evaluation datasets composed of scientific papers and found that DoCollapse outperforms state-of-the-art methods. Results show that simplifying a document graph by homology-preserving topological collapse benefits keyphrase extraction.}, booktitle={2016 IEEE 17th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)}, publisher={IEEE}, author={Guan, Hui and Tang, Wen and Krim, Hamid and Keiser, James and Rindos, Andrew and Sazdanovic, Radmila}, year={2016}, month={Jul} } @article{jablan_sazdanovic_2016, title={From Conway Notation to LinKnot}, volume={670}, ISBN={["978-1-4704-2257-8"]}, ISSN={["1098-3627"]}, DOI={10.1090/conm/670/13445}, journal={KNOT THEORY AND ITS APPLICATIONS}, publisher={American Mathematical Society}, author={Jablan, Slavik V. and Sazdanovic, Radmila}, year={2016}, pages={63–92} } @article{radovic_gerdes_jablan_sazdanovic_2016, title={Plaited polyhedra: A knot theory point of view}, volume={25}, ISSN={["1793-6527"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84980383623&partnerID=MN8TOARS}, DOI={10.1142/s0218216516410066}, abstractNote={Plaited polyhedra, discovered by P. Gerdes in African dance rattle capsules are analyzed from the knot-theory point of view. Every plaited polyhedron can be derived from a knot or link diagram as its dual. In the attempt to classify obtained plaited polyhedra, we propose different methods based on families of knots and links in Conway notation or their corresponding braid families, both leading to the notion of families of plaited polyhedra.}, number={9}, journal={JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS}, publisher={World Scientific Pub Co Pte Lt}, author={Radovic, Ljiljana and Gerdes, Paulus and Jablan, Slavik and Sazdanovic, Radmila}, year={2016}, month={Aug} } @article{kauffman_lambropoulou_sazdanovic_2016, title={Remembering Slavik Jablan}, volume={25}, ISSN={["1793-6527"]}, DOI={10.1142/s0218216516020028}, abstractNote={Journal of Knot Theory and Its RamificationsVol. 25, No. 09, 1602002 (2016) Special Issue for Slavic Jablan; Guest Editors: L. H. Kauffman, R. Sazdanovic and S. LambropoulouNo AccessRemembering Slavik JablanLouis H. Kauffman, Sofia Lambropoulou, and Radmila SazdanovicLouis H. KauffmanUniversity of Illinois at Chicago, USA, Sofia LambropoulouNational Technology University of Athens, Greece, and Radmila SazdanovicNorth Carolina State University, USAhttps://doi.org/10.1142/S0218216516020028Cited by:0 Next AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail References 1. S. Jablan, A new method of generating plane groups of simple and multiple antisymmetry, Acta Cryst. A 42 (1986) 209–212. Crossref, ISI, Google Scholar2. S. Jablan, A new method of deriving and cataloguing simple and Multiple G3l space groups, Acta Cryst. A 43 (1987) 326–337. Crossref, ISI, Google Scholar3. S. Jablan, Algebra of antisymmetric characteristics, Publ. Inst. Math. 47(61) (1990) 39–55. Google Scholar4. S. Jablan, Mackay groups, Acta Cryst. A 49 (1993) 132–137. Crossref, ISI, Google Scholar5. S. Jablan and A. F. Palistrant, Space groups of simple and multiple colored antisymmetry, kristallografiya 38(2) (1993) 4–11. Google Scholar6. S. Jablan, Edge-bicolorings of regular polyhedra, Z. Kristall 210 (1995) 173–176. Crossref, Google Scholar7. S. V. Jablan, Geometry of links, Novi Sad J. Math. 29(3) (1999) 121–139. Google Scholar8. S. V. Jablan, New knot tables, Filomat (Nis), (2002), 141–152, http://www.mi.sanu. ac.rs/jablans/knotab/index.html. Google Scholar9. S. Jablan and R. Sazdanovic, Unlinking number and unlinking gap, J. Knot Theory Ramification 16(10) (2007) 1331–1355. Link, ISI, Google Scholar10. S. Jablan and R. Sazdanovic, Braid family representatives, J. Knot Theory Ramification 17(7) (2008) 817–833. Link, ISI, Google Scholar11. S. Jablan, Lj. Radović and R. Sazdanovic, Tutte and jones polynomials of link families, Filomat 24(3) (2010) 19–33. Crossref, ISI, Google Scholar12. S. Jablan, Lj. Radović and R. Sazdanovic, Adequacy of link families, Publ. Inst. Math. 88(102) (2010) 21–52. Crossref, Google Scholar13. S. Jablan, Lj. Radović and R. Sazdanovic, Pyramidal knots and links and their invariants, MATCH-Commun. Math. Comput. Chem. 65(3) (2011) 541–580. ISI, Google Scholar14. S. Jablan, Lj. Radović and R. Sazdanovic, Knots and links derived from prismatic graphs, MATCH-Commun. Math. Comput. Chem. 66(1) (2011) 65–92. ISI, Google Scholar15. S. Jablan and Lj. Radović, Do you like paleolithic Op-Art? Kybernetes 40(7–8) (2011) 1045–1054 Crossref, ISI, Google Scholar16. S. Jablan, Lj. Radović and R. Sazdanovic, Tute and Jones polynomials of links, polyominoes and graphical recombination patterns, J. Math. Chem. 49(1) (2011) 79–94. Crossref, ISI, Google Scholar17. S. Jablan, Lj. Radović and R. Sazdanovic, Nonplanar graphs derived from gauss codes of virtual knots and links, J. Math. Chem. 49(10) (2011) 2250–2267. Crossref, ISI, Google Scholar18. S. Jablan, Lj. Radović, R. Sazdanovic and A. Zekovic, Mirror-curves and knot mosaics, Comput. Math. Appl. 64(4) (2012) 527–543. Crossref, ISI, Google Scholar19. L. Kauffman, S. Jablan, Lj. Radović and R. Sazdanovic, Reduced relative tutte, Kauffman bracket and Jones polynomials of virtual link families, J. Knot Theory Ramifications 22(4) (2013). Link, ISI, Google Scholar20. A. Henrich, R. Hoberg, S. Jablan, L. Johnson, E. Minten and Lj. Radović, The theory of pseudoknots, J. Knot Theory Ramifications 22(7) (2013). Link, ISI, Google Scholar21. S. Jablan, L. H. Kauffman and P. Lopes, On the maximum number of colors for links, J. Knot Theory Ramifications, 22(3) (2013). Link, ISI, Google Scholar22. L. H. Kauffman and S. V. Jablan, A note on amphicheiral alternating links, J. Knot Theory Ramifications, 22(1) (2013). Link, ISI, Google Scholar23. S. Jablan, Lj. Radović, Unknotting numbers of alternating knot and link families, Publ. Inst. Math. 95(109) (2014) 87–99. Crossref, Google Scholar24. A. Henrich and S. V. Jablan, On the coloring of Pseudoknots, J. Knot Theory Ramifications 23(12) (2014). Link, ISI, Google Scholar25. Lj. Radović and S. Jablan, Meander knots and links, Filomat 29(10) (2015) 2381–2392. Crossref, ISI, Google Scholar26. S. Jablan, L. H. Kauffman and P. Lopes, The Delunification process and minimal diagrams, Topology Appl. 193 (2015) 270–289. Crossref, Google Scholar FiguresReferencesRelatedDetails Recommended Vol. 25, No. 09 Metrics History PDF download}, number={9}, journal={JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS}, author={Kauffman, Louis H. and Lambropoulou, Sofia and Sazdanovic, Radmila}, year={2016}, month={Aug} } @article{sazdanovic_kauffman_radovic_2016, title={Remembrance Slavik Jablan (1952-2015)}, volume={25}, ISSN={["1793-6527"]}, DOI={10.1142/s021821651602003x}, number={9}, journal={JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS}, author={Sazdanovic, Radmila and Kauffman, Louis H. and Radovic, Ljiljana}, year={2016}, month={Aug} } @article{zekovic_jablan_kauffman_sazdanovic_stosic_2016, title={Unknotting and maximum unknotting numbers}, volume={25}, ISSN={["1793-6527"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84979240749&partnerID=MN8TOARS}, DOI={10.1142/s0218216516410108}, abstractNote={We introduce concepts of the maximum unknotting number and the mixed unknotting number, taking into consideration the Bernhard–Jablan Conjecture about computing the unknotting number based only on minimal knot diagrams. The existence of Kauffman knots (alternating knots, such that a crossing change does not change their minimal crossing number) was first suggested by Kauffman. We extend the concept and offer three related classes of knots named: Kauffman knots, Zeković knots and Taniyama knots.}, number={9}, journal={JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS}, publisher={World Scientific Pub Co Pte Lt}, author={Zekovic, Ana and Jablan, Slavik and Kauffman, Louis and Sazdanovic, Radmila and Stosic, Marko}, year={2016}, month={Aug} } @inproceedings{khovanov_sazdanovic_2015, title={A Categorification of One-Variable Polynomials}, booktitle={Discrete Mathematics and Theoretical Computer Science}, author={Khovanov, M. and Sazdanovic, R.}, year={2015}, pages={937–948} } @inproceedings{sazdanovic_yip_2015, title={A categorification of the chromatic symmetric polynomial}, booktitle={Discrete Mathematics and Theoretical Computer Science}, author={Sazdanovic, R. and Yip, M.}, year={2015}, pages={631–642} } @article{khovanov_sazdanovic_2015, title={Categorifications of the polynomial ring}, volume={230}, ISSN={["1730-6329"]}, DOI={10.4064/fm230-3-3}, abstractNote={We develop a diagrammatic categorification of the polynomial ring ${\mathbb {Z}}[x]$. Our categorification satisfies a version of Bernstein–Gelfand–Gelfand reciprocity property with the indecomposable projective modules corresponding to $x^n$}, number={3}, journal={FUNDAMENTA MATHEMATICAE}, publisher={Institute of Mathematics, Polish Academy of Sciences}, author={Khovanov, Mikhail and Sazdanovic, Radmila}, year={2015}, pages={251–280} } @article{crowe_darvas_huylebrouck_kappraff_kauffman_lambropoulou_przytycki_radovic_sazdanovic_spinadel_et al._2015, title={In Memoriam: Slavik Jablan 1952-2015 Obituary}, volume={7}, ISSN={["2073-8994"]}, DOI={10.3390/sym7031261}, abstractNote={first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing: Column Width: Background: Open AccessObituary In Memoriam: Slavik Jablan 1952–2015 by Donald Crowe 1, György Darvas 2, Dirk Huylebrouck 3, Jay Kappraff 4, Louis Kauffman 5, Sofia Lambropoulou 6, Jozef Przytycki 7, Ljiljana Radović 8, Radmila Sazdanovic 9,*, Vera W. De Spinadel 10, Ana Zeković 11 and Symmetry Editorial Office 12 1 Department of Mathematics, University of Wisconsin-Madison, Madison WI 53706, USA 2 Symmetrion, 29 Eötvös St., Budapest H-1067, Hungary 3 Faculty of Architecture, KU Leuven, Hoogstraat 51, 9000 Gent, Belgium 4 Department of Mathematical Sciences, New Jersey Institute of Technology University Heights, Newark, NJ 07102, USA 5 Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA 6 Department of Applied Mathematics, National Technical University of Athens, Athens 15780, Greece 7 Departments of Mathematics, The George Washington University, Monroe Hall, Room 240, 2115 G St. NW., Washington, DC 20052, USA 8 Faculty of Mechanical Engineering, Aleksandra Medvedeva, Niš 18000, Serbia 9 Department of Mathematics, North Carolina State University, PO Box 8205, Raleigh, NC 27695, USA 10 Jose M. Paz 1131 Florida (1602) Provincia de Buenos Aires, Argentina 11 Department of Mathematical Sciences, Faculty of Technology and Metallurgy, Karnegijeva 4, Belgrade 11000, Serbia 12 Klybeckstrasse 64, Basel 4057, Switzerland add Show full affiliation list remove Hide full affiliation list * Author to whom correspondence should be addressed. Symmetry 2015, 7(3), 1261-1274; https://doi.org/10.3390/sym7031261 Received: 29 May 2015 / Revised: 29 May 2015 / Accepted: 1 June 2015 / Published: 15 July 2015 Download Download PDF Download PDF with Cover Download XML Download Epub Browse Figures Versions Notes Graphical Abstract 1. In Memory of Slavik Jablan by Ljiljana Radovic and Radmila SazdanovicAfter a long and brave battle with a serious illness, our dear friend and colleague Slavik Jablan passed away on 26 February 2015. The world is deprived of a remarkable mathematician, a great artist, a wonderful man and a dear friend. He made significant contributions to many areas of mathematics: geometry, group theory, mathematical crystallography, the theory of symmetry, antisymmetry, colored symmetry, combinatorial geometry, knot theory, visual mathematics and mathematical art.Slavik Jablan was born on 10 June 1952 in Sarajevo, Yugoslavia (now in Bosnia-Herzegovina). He grew up in Sarajevo, Dubrovnik and Belgrade. He studied at the University of Belgrade and graduated in 1977 in theoretical mathematics from the Faculty of Mathematics, University of Belgrade, where he also obtained his MA in 1981 and PhD in 1984, with a thesis entitled Theory of Simple and Multiple Antisymmetry in E 2 and E 2 ∖ { O } . In 1985 and 1988, he pursued an advanced scientific program in colored symmetry at the University of Kishinev in the former USSR (now Chişinău, Republic of Moldova). He also held numerous visiting positions, including in the USA and Canada (1990): the University of Wisconsin-Madison; Indiana University Bloomington, Department of Anthropology; Tsukuba University, Tsukuba, Science City, Japan (1999); Fulbright Scholar, USA (2003–2004); and many others.Slavik Jablan started his career at the PTT School center and at the Pedagogical Academy for Teacher Training in Belgrade, where he worked until 1984. He moved to the Faculty of Philosophy, Department of Mathematics at the University of Niš in southern Serbia, where he was a professor of geometry. In 1999, he returned to Belgrade, to work at the College of Information and Communication Technology. He designed and taught a visual mathematics course for designers at the Metropolitan University in Belgrade. Over the years, he was a Researchmember of the Mathematical Institute of the Serbian Academy of Science and Arts, the Editor of the VisMath electronic journal (http://www.mi.sanu.ac.rs/vismath) and the Editor in Chief of Symmetry. He was a member of the Advisory Board of The International Society for the Interdisciplinary Study of Symmetry, as well as a member of many other math societies.His scientific roots lie in deriving and cataloging groups of simple and multiple antisymmetry based on an antisymmetry characteristic (AC) method that he developed in his PhD thesis. Using AC methods, it was possible to derive and to distinguish different antisymmetry groups based on their antisymmetry characteristic. He was also interested in enantiomorphism forms and chirality. His first PhD student Ljiljana Radovic continued this work and implemented the method of antisymmetry characteristic in the computation of multiplied antisymmetry groups by computer. He published more than 30 scientific papers on this topic, as well as several monographs: Theory of Symmetry and Ornament, APXAIA, Belgrade, 1984 (in Serbo-Croatian); Geometry in Pre-Scientific Period & Ornament Today, Math. Inst., History of Math. and Mech. Sci., 3, Belgrade, 1989; Theory of Symmetry and Ornament, Math. Inst., Belgrade, 1995 (http://www.mi.sanu.ac.rs/jablans/mon.htm); Symmetry, Ornament and Modularity, World Scientific, New York, 2002.Starting in the mid-1990s, Slavik Jablan expanded his interests to knot theory, an area of low-dimensional topology. His unique background enabled Slavik to discover various patterns in the world of knots and their invariants. In 1995, he proposed the so-called Bernhard–Jablan conjecture about an invariant called the unknotting number. His idea of utilizing Conway notation as well as knot and link families is the core of the knot theory software package LinKnot that he developed together with Radmila Sazdanovic. They also published a book titled LinKnot: Knot Theory by Computer (World Scientific, Singapore, 2007) along with a webMathematica version of LinKnot (http://math.ict.edu.rs/). Slavik published many significant scientific papers with theoretical and experimental results, and influenced the area of knot theory through collaborations and inspiring students to take up this field of study. Till the very end, Slavik was working with his last PhD student, Ana Zekovic, who defended her PhD thesis “Conway notation and its appliance in knot distance determination methods, in knot theory” on the same day when Slavik Jablan passed away.Slavik Jablan was one of the leading experts in visual mathematics. All his life, he was building bridges between science and art. He was interested in the history of ornamental art, patterns, modularity, visual perception, Celtic art, ornamental design, key-patterns, Roman mazes and labyrinths, Paleolithic ornaments and op art puzzles. His contribution to the symmetry approach of ornamental design by SpaceTiles, KnotTiles and especially OpTiles based on modularity will be remembered by all. Slavik was an accomplished painter and math artist with more than 15 solo exhibitions, and his work on Two-Colored Ornamental Tilings based on modularity and Mathematics and Design (1998) winning one of the awards at the International Competition of Industrial Design and New Technology CEVISAMA1987 (Valencia, Spain).He was an avid promoter of the concept of visual geometry and visual mathematics, with emphasis on symmetry in visual and ornamental art, at numerous conferences, such as ISIS-Symmetry, BRIDGES, Gathering for Gardner, as well as in the series of lectures all around the world: at the University of Wisconsin (Madison, Department of Mathematics), Indiana University (Bloomington, Department of Anthropology), Technische Universität (Vienna, Austria, Department of Geometry), Symmetry and Visual Perception at Faculty of Philosophy (Belgrade, Department of Experimental Psychology), Symmetry and Visual Arts at the National Museum (Belgrade), Classification of Ornaments at the Museum of Contemporary Art (Belgrade). His love and passion for ornamental art lasted all his life. He was one of the founders of the Experience Workshop Math-Art Movement and creator of the course “Visual Mathematics and Design” at the Faculty of Information Technologies (Belgrade), and his ideas and contributions were essential for the success of the Tempus Project on Visual Mathematics. He was invited to give the contribution “Classification of Ornaments” for the catalog of the exhibition “Memory Update: Ornaments of Serbian Medieval Frescoes”, held at the Museum of Applied Arts, Belgrade, November 2013–March 2014. In November 2014, he had his last solo exhibition, “Do you like Paleolithic ornamental art?” in the Radio Television of Serbia Gallery in Belgrade. Figure 1. Slavik’s lecture at the Summer Institute in Eger, Hungary, July 2013. Figure 1. Slavik’s lecture at the Summer Institute in Eger, Hungary, July 2013. Figure 2. Knot-2013 ICTS, Tata Institute for Fundamental Research, India, December 2013. Figure 2. Knot-2013 ICTS, Tata Institute for Fundamental Research, India, December 2013. Slavik Jablan was a brilliant mathematician, artist and, above all, a wonderful man and a devoted friend, and he will be missed and remembered by all. 2. Donald Crowe, Professor Emeritus, University of Wisconsin-Madison, WI, USASlavik Jablan was born 1952 in Sarajevo at a time midway between the two dramatic events by which Sarajevo is otherwise known to the outside world: the assassination of Archduke Ferdinand, which set off World War I, and the siege of Sarajevo, which marked the disintegration of modern Yugoslavia. Slavik emerged from this second event to become a prominent spokesman and practitioner for symmetry in its mathematical and artistic forms. In the early 1990s, he received a grant to travel to America to become acquainted with prominent figures, such as H.S.M. Coxeter (who had recently helped organize an Escher conference in Rome) and Arthur Loeb, whose book on color symmetry dealt with a topic much like Slavik’s own 1984 Belgrade thesis (published in English as Theory of Symmetry and Ornament, 1995).Growing out of this trip, he received a Fulbright scholarship to return to America to work with me and others on real-world symmetry. However, this was suspended when the war in Yugoslavia intervened. For the next few years, he made many contacts in Europe, especially via the ISIS-Symmetry organization of Georgy Darvas and Denes Nagy. By 1998, he founded the electronic journal VisMath, where “visual mathematics” came to include not only traditional geometric topics, but decorative art (at which Slavik was an expert) and symmetries of primitive and ethnographic art (providing an outlet, for example, for the work of Paulus Gerdes, who sadly died a few months before Slavik). I was honored to have him accept a small paper of mine to be the first paper in the first issue, and as editor, he enlivened the paper, so that it lights up like a shopping mall. Slavik’s own art can be seen in several papers in the MathArt section of VisMath. The Bridges organization was founded in America about the same time, and Slavik became an early participant in its annual conferences, including editing one of the proceedings. At the Bridges conference that year, he entertained some of the participants by producing almost instant watercolors, including marine scenes from the Adriatic coast of his homeland. Each New Years, he produced modular or op art calendars. Not until 2004 was he able to get his 1992 Fulbright reinstated, and he used the time in Wisconsin to complete his knot theory book (published with Radmila Sazdanovic) and work with Jay Kappraff in New Jersey.At the time of his death, Slavik was the editor not only of VisMath, but of the electronic journal Symmetry published in Switzerland. It will be hard to fill his niche in the world of mathematics and art, impossible to fill it in the memories of those who were his friends. 3. György Darvas, Editor of Symmetry: Culture and Science, Budapest, HungaryI had known Slavik since 1988, and we met in the following year. Twenty seven years is a long time: uncountable correspondence, several common projects, many personal meetings. Our relation was not restricted to collaboration in science. We visited each other, and when his home country was the subject of bombing by foreign armed forces, my wife and I invited him and his kind wife Jadranka to survive the hard period in our home in Budapest. Although they did not want to leave their apartment unattended in that period, they did not forget it, even until our last meeting last autumn. Of course, there were less and more intensive periods in the contact over such a long time, but we could recover our collaboration when our mutually beloved common theme, symmetry, demanded it. The last few years belonged again to collaborative periods, and we had further plans that, unfortunately, will never be realized. I appreciated greatly his scholarly intellect, his wide knowledge both in the sciences and the arts and his smart attitude. Figure 3. Slavik Jablan at the Bridges Conference in Pecs, Hungary, 2010. Figure 3. Slavik Jablan at the Bridges Conference in Pecs, Hungary, 2010. 4. Dirk Huylebrouck, Faculty of Architecture, KU Leuven, Gent, BelgiumI first met Slavik Jablan somewhere in the middle of the 1990s during a mathematics and design conference in San Sebastian, Spain, where his lecture was elected “best lecture of the meeting”. We would keep in touch, through many conferences and uncountable emails during the years. Slavik came to Belgium several times, and I always programmed him as the first speaker of my conferences on “Mathematics and Art”, to make sure to have a good start.As a foremost member of many mathematics and art societies, he started the very first “Visual Mathematics” website, at times when making and spreading Internet sites was not as evident as it is now. It became a top virtual library on mathematics and art. I stand behind the point of view that it was Slavik who coined the term “visual mathematics”, though I know he found this statement exaggerated. In 2012, he gave me a copy of a chapter of an unusual course, entitled “visual mathematics”, which he was teaching to graphical designers at the Belgrade Metropolitan University. I tried to teach that chapter to my own architecture students in Belgium and I can confirm firsthand that it would be a pity if this course were to not be published nor taught in the future. It was Slavik’s opinion that designers and artists have the obligation and right to know mathematics, and that was right.Over the years, we discussed many times how to find some support for those visual mathematics activities. Several draft project versions were written, rejected for all kinds of reasons, sometimes related to the political situation of Serbia. However, Slavik persisted, and in the end, we got a Tempus project approved on visual mathematics. Symbolically, the project finished in 2014, as if he had waited for it to be completed. We will not forget how much he enjoyed the attention of the general public for mathematics, during the “Belgrade Summer School on Visual Mathematics”, in 2014, at the shores of the River Danube. Slavik was the foremost mathematician in the group, and that was right. Figure 4. At the Math and Art meeting in Ghent, Belgium, 20–21 May 2011. Figure 4. At the Math and Art meeting in Ghent, Belgium, 20–21 May 2011. After a talk of mine in Israel somewhere in 1998 about the oldest mathematical object, the Congolese Ishango rod, we engaged in a discussion about this object. We did not entirely agree, but on the contrary, that was not a problem. Later, in 2007, he would even come to my conference “Ishango, 22000 and 50 years later: the cradle of mathematics?” at the Flemish Royal Academy of Belgium. Additionally, in 2013, a contribution by his hand, entitled “A Second Opinion on the Ishango Rod”, would be a part of a chapter in my book “Belgium + Mathematics”. This genuine interest he had for Africa was also confirmed in a kind of expedition we held in Buenos Aires, Argentina. Someone informed us that an Argentine citizen had a large collection of Congolese fabrics that he did not exactly know what to do with and how they should be grouped together, and for some reason, he did not want to follow the official steps. Yet, we went to visit him, took numerous pictures and crowned the expedition with a paper on a group theory classification of Congolese fabrics in Buenos Aires. Mathematics combined with adventure, and that was fun.Because of that Tempus project, I finally had an official reason to visit Slavik in Belgrade, in April 2013. I thought it would be awkward, since, after all, his Serbia had been heavily bombed by NATO, in 1999. He reported about the bombing to me from his side, while I was e-mailing from Brussels, the city of NATO’s head quarters. Still, the welcome in Belgrade could not be better, and I have hardly ever been in a country as hospitable as Serbia. Additionally, even posthumously, he continues to spread the message about the mathematical beauty of Serbia, through the exhibition “Memory Update: Ornaments of Serbian Medieval Frescoes”. This exhibition, to which he contributed the mathematical aspects, is starting a tour around the world. His spirit will travel with it. 5. Jay Kappraff, New Jersey Institute of Technology, Newark, NJ, USASlavik was a prince among mathematicians. I had a long and fruitful history with him and also with his wonderful wife Jadranka and son Ivan. In 2005, he got a Fulbright, and we spent three months together at NJIT with Slavik living in my town of South Orange. During this time, we wrote a couple of papers together and had many adventures. When it came time to part, he told me that he wanted to give me a gift. At the end of the weekend, he presented me with two large watercolor paintings. Being overwhelmed by my own work, I put them in my attic. It was three years later that I took a careful look at them. They were masterpieces, and they now adorn my apartment. One in particular is priceless. It keeps another of his still life paintings company in a place of honor in my living room. For this past year, we have been working on a book to help faculty who wish to teach a course in Math of Design for which we have contracted with World Scientific. Although he cannot complete this task, he has bequeathed it to two former students of his, Ljiljana Radovic and Radmila Sazdanovic. They will carry on this work, which will be a fitting tribute to Slavik. In 2008, I paid a visit to Slavik in Belgrade. Ivan, Slavik’s son, who is an archivist of Serbian history and culture, took me on a historical tour of Belgrade, while Jadranka shared her novel approach to musical theory with me. Ivan is a fine chorale singer in the Serbian choral tradition. It so happened that in 2012, my daughter-in-law was visiting Belgrade as a choral conductor during which Slavik showed her great hospitality. Slavik was broad gauged in his approach to art and life with strong interests in both music, art and design. He also had a passion for the history of design going back as far as 23,000 BC to the Mezin culture. He collected designs through all of Europe, but particularly in Eastern Europe. His essay on mathematics in the art of Salvador Dali, never published, is a masterpiece. In recent years, he became one of the world’s authorities on knot theory, establishing a fruitful collaboration with Prof. Louis Kauffman, the leading authority on this subject, and he continued, almost single-handedly, to publish the on-line Journal of Visual Mathematics. In fact, he did all of this in a lifetime full of physical challenges. There was nothing that could deter him and disturb his laser beam concentration and focus. However, with all of his adversities, he never once complained. It all became part of his special dance through life. Even through his final disease, his focus was more on sharing yet more of his extensive oeuvre than on the excruciating pain he was enduring. This world is richer for his presence and poorer now that he has departed. 6. Louis H. Kauffman, University of Illinois at Chicago, Chicago, IL, USAI first met Slavik Jablan in the 1990s, and it was clear from the start that he had a new and different point of view about knot theory and indeed a different and artistic view of mathematics in general. He had worked in symmetry, ornaments, painting and design before coming to knot theory, and his vision of knot theory was suffused with the rich appearance of knotting and weaving in tapestry, painting and design, the use of knots both practical and decorative in rope-work and for ships and construction that extended back into antiquity. While for the rest of us, knot theory was a modern phenomenon that began in the 19th century, for Slavik, it was a mathematical phenomenon that had begun in the roots of civilization. I say a mathematical phenomenon, because it was mathematics in the patterns and calculations of weaving and painting and design. Knot theory became a chapter of systematized, algebraic and abstract mathematics only recently, but it had been part of a tradition that respected pattern and craft and practical calculation for long before that. This meant that Slavik’s view of knot theory was wider, more artistic and, at the same time, more computational than the rest of us. He was genuinely interested in how knots and links could fall into visual and recursively generated families. He was sensitive to the extraordinary combination of calculation and topological patterning that lay behind the abstract definitions that most of us explore. Consequently, he was willing to enter into computational investigations and conjectures about knots and links from all areas of the theory. He never stopped examining, from his point of view, any statement of relationships. This led him to make very fine conjectures (such as the Bernhard–Jablan conjecture about the unknotting number), and it made him an absolutely extraordinary collaborator. One could invent or discover some combinatorial property of knots and, the next day, send Slavik an account of the definition and ask some question about generating examples of a type or whether such and such a phenomenon might occur. He would immediately understand and put the matter to his computing system (LinKnot) and generate many examples, and the work and the conjecturing would go forward. He had a completely open mind, full creativity, devotion to art and mathematics, love of collaboration and love of friends and humanity. He was a person of value beyond words. We are going to miss him more and more as time goes on. Figure 5. Advanced School and Conference on Knot Theory and its Applications to Physics and Biology, International Centre for Theoretical Physics (ICTP), Trieste, Italy, 11–29 May 2009. Figure 5. Advanced School and Conference on Knot Theory and its Applications to Physics and Biology, International Centre for Theoretical Physics (ICTP), Trieste, Italy, 11–29 May 2009. 7. Sofia Lambropoulou, National Technical University of Athens, Athens, GreeceI first talked with Slavik Jablan at the International Centre for Theoretical Physics (ICTP) in Trieste in May 2009, where we co-organized, together with Louis Kauffman and Jozef Przytycki, the “Advanced School and Conference on Knot Theory and its Applications to Physics and Biology”. In the period prior to the meeting, we had discussed, over email, being the two local organizers. However, I had already, long before, heard of him and his ability to do programming and computations with knots, which would lead him to formulating deep conjectures.At the ICTP, every evening of the meeting, we would sit with the four organizers at the terrace of the dinning cafeteria over a pitcher of wine, and we would see to organizational matters that needed attention. One evening, many of us played, on the spot, as actors at the improvised play “Dehn’s dilemma”. This was an idea I had been amusing myself with for a while, imagining topological surgery as medical surgery, of which I talked to Cameron Gordon early that evening with the pressing request that we do some kind of play. Cameron soon started forming a scenario in his mind, involving rational and irrational surgery and Dr. Dehn being an absurd doctor, which evolved into a superb, unforgettable comedy in the mind and hands of Colin Adams: “Unhappy solid torus wishes to become lens space, consults mad doctor who performs irrational surgery. The love interest of the torus falls for the figure 8 knot complement, and high drama occurs.” From YouTube: https://www.youtube.com/watch?v=UM-U5RE2mN0. I recall that Slavik enjoyed his role enormously, I think, being the solid torus, as well as the whole event. In September 2010, we co-organized the same team “Knots in Chicago”. Figure 6. Slavik Jablan and Jesus Juyumaya in Ancient Olympia, Greece, 2013. Figure 6. Slavik Jablan and Jesus Juyumaya in Ancient Olympia, Greece, 2013. Two years later, I was asking him if he would be willing to help with a puzzling problem: with my collaborator Jesus Juyumaya from Chile, we had constructed an infinitum of knot and link invariants via the Yokonuma–Hecke algebras, which are quotients of the framed braid group over a complex quadratic relation. These invariants needed to be compared with known ones, especially with the Homflypt polynomial, but a direct comparison was not obvious. My student Konstantinos Karvounis and Michael Chmutov had created computational packages, so we started making computations (together with Sergei Chmutov). Slavik provided us immediately with several pairs of knots and links in braid form sharing the same Homflypt polynomial. We could quickly come to the idea that our invariants were no stronger than the Homflypt polynomial, but a theoretical proof was not in sight.In April 2013, I visited him in Belgrade in order to look at the problem more systematically. There, I had the opportunity to meet his wife Jadranka, a music teacher, a wonderful person and a great cook. Their apartment, which had an artistic bohemian air, was full of Slavik’s amazing oil paintings and several rare and interesting artistic objects.We started skyping with Konstantinos and making testing computations on knot and link families that Slavik had indicated. Soon, he was able to figure out some pattern. In September 2013, Slavik and Jadranka visited me in Athens, where, together with Konstantinos and Jesus, we managed to formulate our main conjecture: on knots, our invariants are topologically equivalent to the Homflypt polynomial. This was proven recently and will appear in a joint paper with his name on it. The conjecture was proven recently, and we even showed that on links, the behavior is different.Slavik has been an excellent, tireless researcher and highly intelligent, with his own unique taste for mathematics and art; a good friend, a person with broad and deep human awareness, honest, open-minded and non-judgmental, in the company of whom one would feel a warm and calm light. 8. Jozef Przytycki, George Washington University, Washington, DC, USAI first met Slavik at Knots in Hellas, Greece 1998. I remember him mostly as a strong proponent of the conjecture about the unknotting number, stating that under certain conditions, the unknotting number can be obtained from a minimal diagram, now known as the Bernhard –Jablan conjecture. The next time we met at Knots in Washington XVII in 2003, when he brought with him his student collaborator Radmila Sazdanovic. Radmila soon became my student, so following the example of Hugo Steinhaus, I can say that “Radmila was one of Slavik’s biggest mathematical discoveries”. One of Slavik’s greatest gifts was his passion of relating mathematics and art. This he advocated whenever I met him: in Trieste in May 2009 or, the last time, in India in December 2013. For all of this, he will be remembered! 9. Ljiljana Radovic, Faculty of Mechanical Engineering, Niš, SerbiaI had the pleasure of being a student of Professor Slavik Jablan at the Faculty of Philosophy, Department of Mathematics at the University of Niš in 1990. Prof. Slavik Jablan also was my mentor for my Masters and PhD theses. During 25 years of collaborations, he became my dear friend. It was a great privilege to learn from him, not only mathematics and geometry, not only about symmetry and knots, but also about arts and artists, paintings and painters, about anthropology, archeology, architecture, ethnology, about movies, music and windsurfing.Slavik Jablan was a brilliant mathematician and artist and, above all, a wonderful man, unselfish in shearing ideas, information, knowledge, books and graphics. It was not always easy to follow his quick mind nor keep track of his numerous ideas, amazing creativity and great intelligence, but it was a great honor to work with him and to be his friend. He always had new ideas about what we could do next and what we should write about and prepared new materials, workshops and projects. During all of these years, we worked on many papers and projects together. In the last two years, we were working on several projects. Our work on the paper “Classification of ornaments based on Serbian fresco medieval art” is especially precious for me, but also our work on the book Visual mathematics with Jay Kapraff, education material for the teachers included in the Tempus project “Visuality & Mathematics: Experimental Education of Mathematics through Visual Arts, Sciences and Playful Activities”, workshops and lectures.We all were able to learn how to live and fight in sp}, number={3}, journal={SYMMETRY-BASEL}, author={Crowe, Donald and Darvas, Gyoergy and Huylebrouck, Dirk and Kappraff, Jay and Kauffman, Louis and Lambropoulou, Sofia and Przytycki, Jozef and Radovic, Ljiljana and Sazdanovic, Radmila and Spinadel, Vera W. and et al.}, year={2015}, month={Sep}, pages={1261–1274} } @article{przytycki_sazdanovic_2014, title={Torsion in Khovanov homology of semi-adequate links}, volume={225}, ISSN={["1730-6329"]}, DOI={10.4064/fm225-1-13}, abstractNote={The goal of this paper is to address A. Shumakovitch's conjecture about the existence of ${\mathbb {Z}}_2$-torsion in Khovanov link homology. We analyze torsion in Khovanov homology of semi-adequate links via chromatic cohomology for graphs, which pr}, number={1}, journal={FUNDAMENTA MATHEMATICAE}, publisher={Institute of Mathematics, Polish Academy of Sciences}, author={Przytycki, Jozef H. and Sazdanovic, Radmila}, year={2014}, pages={277–303} } @article{kauffman_jablan_radović_sazdanović_2013, title={REDUCED RELATIVE TUTTE, KAUFFMAN BRACKET AND JONES POLYNOMIALS OF VIRTUAL LINK FAMILIES}, volume={22}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84877608644&partnerID=MN8TOARS}, DOI={10.1142/s0218216513400038}, abstractNote={This paper contains general formulae for the reduced relative Tutte, Kauffman bracket and Jones polynomials of families of virtual knots and links given in Conway notation and discussion of a counterexample to the Z-move conjecture of Fenn, Kauffman and Manturov.}, number={04}, journal={Journal of Knot Theory and Its Ramifications}, publisher={World Scientific Pub Co Pte Lt}, author={KAUFFMAN, LOUIS H. and JABLAN, SLAVIK and RADOVIĆ, LJILJANA and SAZDANOVIĆ, RADMILA}, year={2013}, month={Apr}, pages={1340003} } @article{nanda_sazdanović_2013, title={Simplicial Models and Topological Inference in Biological Systems}, DOI={10.1007/978-3-642-40193-0_6}, abstractNote={This article is a user’s guide to algebraic topological methods for data analysis with a particular focus on applications to datasets arising in experimental biology. We begin with the combinatorics and geometry of simplicial complexes and outline the standard techniques for imposing filtered simplicial structures on a general class of datasets. From these structures, one computes topological statistics of the original data via the algebraic theory of (persistent) homology. These statistics are shown to be computable and robust measures of the shape underlying a dataset. Finally, we showcase some appealing instances of topology-driven inference in biological settings, from the detection of a new type of breast cancer to the analysis of various neural structures.}, journal={Discrete and Topological Models in Molecular Biology}, publisher={Springer Berlin Heidelberg}, author={Nanda, Vidit and Sazdanović, Radmila}, year={2013}, month={Oct}, pages={109–141} } @inproceedings{jablan_sazdanovic_2012, place={Singapore}, series={Series on Knots and Everything}, title={Diagrammatic knot properties and invariants}, booktitle={Introductory lectures on knot theory : selected lectures presented at the Advanced School and Conference on Knot Theory and Its Applications to Physics and Biology, ICTP, Trieste, Italy, 11-29 May 2009}, publisher={World Scientific}, author={Jablan, S. and Sazdanovic, R.}, year={2012}, pages={162–186}, collection={Series on Knots and Everything} } @article{sazdanovic_2012, title={Diagrammatics in Art and Mathematics}, volume={4}, ISSN={2073-8994}, url={http://dx.doi.org/10.3390/sym4020285}, DOI={10.3390/sym4020285}, abstractNote={This paper explores two-way relations between visualizations in mathematics and mathematical art, as well as art in general. A collection of vignettes illustrates connection points, including visualizing higher dimensions, tessellations, knots and links, plotting zeros of polynomials, and new and rapidly developing mathematical discipline, diagrammatic categorification.}, number={2}, journal={Symmetry}, publisher={MDPI AG}, author={Sazdanovic, Radmila}, year={2012}, month={May}, pages={285–301} } @book{sazdanovic_barallo_budin_durity_fenyvesi_jablan_takacs_radovic_stettner_2012, place={Kaposvar}, title={Experience-centered approach and Visuality in the Education of Mathematics and Physics}, ISBN={978-963-9821-52-1}, publisher={Kaposvar University}, year={2012} } @inproceedings{sazdanovic_2012, title={Fisheye View of Tessellations}, booktitle={Bridges: Mathematical Connections in Art, Music and Science}, author={Sazdanovic, R.}, year={2012}, pages={361–364} } @article{baranovsky_sazdanovic_2012, title={Graph homology and graph configuration spaces}, volume={7}, ISSN={2193-8407 1512-2891}, url={http://dx.doi.org/10.1007/S40062-012-0006-3}, DOI={10.1007/S40062-012-0006-3}, abstractNote={If R is a commutative ring, M a compact R-oriented manifold and G a finite graph without loops or multiple edges, we consider the graph configuration space M G and a Bendersky–Gitler type spectral sequence converging to the homology H *(M G , R). We show that its E 1 term is given by the graph cohomology complex C A (G) of the graded commutative algebra A = H*(M, R) and its higher differentials are obtained from the Massey products of A, as conjectured by Bendersky and Gitler for the case of a complete graph G. Similar results apply to the spectral sequence constructed from an arbitrary finite graph G and a graded commutative DG algebra $${\mathcal{A}}$$ .}, number={2}, journal={Journal of Homotopy and Related Structures}, publisher={Springer Science and Business Media LLC}, author={Baranovsky, Vladimir and Sazdanovic, Radmila}, year={2012}, month={Apr}, pages={223–235} } @article{jablan_radović_sazdanović_2012, title={Knots and links in architecture}, volume={7}, ISSN={1788-1994 1788-3911}, url={http://dx.doi.org/10.1556/pollack.7.2012.s.6}, DOI={10.1556/pollack.7.2012.s.6}, abstractNote={This paper contains a survey of different methods for generating knots and links based on geometric polyhedra, suitable for applications in architecture chemistry, biology (or even jewelry). We describe several ways of obtaining 4-valent polyhedral graphs and their corresponding knots and links from basic polyhedra: mid-edge construction, cross-curve and double-line covering, and edge doubling construction. These methods are implemented in the Mathematica-based program LinKnot and can be applied to the data base of basic polyhedra. In a similar way, an edge doubling construction transforms fullerene graphs into alternating knot and link diagrams. In the last part of the paper is proposed the use of virtual knots and links and the corresponding nonplanar graphs obtained from their Gauss codes.}, number={Supplement 1}, journal={Pollack Periodica}, publisher={Akademiai Kiado Zrt.}, author={Jablan, Slavik and Radović, Ljiljana and Sazdanović, Radmila}, year={2012}, month={Jan}, pages={65–76} } @article{jablan_radović_sazdanović_zeković_2012, title={Knots in Art}, volume={4}, ISSN={2073-8994}, url={http://dx.doi.org/10.3390/sym4020302}, DOI={10.3390/sym4020302}, abstractNote={We analyze applications of knots and links in the Ancient art, beginning from Babylonian, Egyptian, Greek, Byzantine and Celtic art. Construction methods used in art are analyzed on the examples of Celtic art and ethnical art of Tchokwe people from Angola or Tamil art, where knots are constructed as mirror-curves. We propose different methods for generating knots and links based on geometric polyhedra, suitable for applications in architecture and sculpture.}, number={2}, journal={Symmetry}, publisher={MDPI AG}, author={Jablan, Slavik and Radović, Ljiljana and Sazdanović, Radmila and Zeković, Ana}, year={2012}, month={Jun}, pages={302–328} } @article{jablan_radović_sazdanović_zeković_2012, title={Mirror-curves and knot mosaics}, volume={64}, ISSN={0898-1221}, url={http://dx.doi.org/10.1016/j.camwa.2011.12.042}, DOI={10.1016/j.camwa.2011.12.042}, abstractNote={Inspired by the paper on quantum knots and knot mosaics (Lomonaco and Kauffman, 2008 [18]) and grid diagrams (or arc presentations), used extensively in the computations of Heegaard–Floer knot homology (Bar-Natan, 0000 [16], Cromwell, 1995 [21], Manolescu et al., 2007 [22]), we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot mosaics (Lomonaco and Kauffman, 2008 [18]), mirror-curves, and grid diagrams (Bar-Natan, 0000 [16], Cromwell, 1995 [21], Kuriya, 2008 [20], Manolescu et al., 2007 [22]). Hence, we introduce codes for mirror-curves treated as knot or link diagrams placed in rectangular square grids, suitable for software implementation. We provide tables of minimal mirror-curve codes for knots and links obtained from rectangular grids of size 3×3 and p×2 (p≤4), and describe an efficient algorithm for computing the Kauffman bracket and L-polynomials (Jablan and Sazdanović, 2007 [8], Kauffman, 2006 [11], Kauffman, 1987 [12]) directly from mirror-curve representations.}, number={4}, journal={Computers & Mathematics with Applications}, publisher={Elsevier BV}, author={Jablan, Slavik and Radović, Ljiljana and Sazdanović, Radmila and Zeković, Ana}, year={2012}, month={Aug}, pages={527–543} } @article{jablan_sazdanović_2011, title={DIAGRAMMATIC KNOT PROPERTIES AND INVARIANTS}, DOI={10.1142/9789814313001_0008}, journal={Introductory Lectures on Knot Theory}, publisher={WORLD SCIENTIFIC}, author={Jablan, S. V. and Sazdanović, R.}, year={2011}, month={Sep}, pages={162–186} } @article{jablan_radovi?_sazdanovi?_2011, title={Knots and links derived from prismatic graphs}, volume={66}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-80155214125&partnerID=MN8TOARS}, number={1}, journal={Match}, author={Jablan, S. and Radovi?, L. and Sazdanovi?, R.}, year={2011}, pages={65–92} } @article{jablan_radović_sazdanović_2011, title={Nonplanar graphs derived from Gauss codes of virtual knots and links}, volume={49}, ISSN={0259-9791 1572-8897}, url={http://dx.doi.org/10.1007/S10910-011-9884-6}, DOI={10.1007/S10910-011-9884-6}, number={10}, journal={Journal of Mathematical Chemistry}, publisher={Springer Science and Business Media LLC}, author={Jablan, Slavik and Radović, Ljiljana and Sazdanović, Radmila}, year={2011}, month={Aug}, pages={2250–2267} } @article{jablan_radovi?_sazdanovi?_2011, title={Pyramidal knots and links and their invariants}, volume={65}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-80051537311&partnerID=MN8TOARS}, number={3}, journal={Match}, author={Jablan, S. and Radovi?, L. and Sazdanovi?, R.}, year={2011}, pages={541–580} } @phdthesis{sazdanović_2010, title={Categorification of Knot and Graph Polynomials and the Polynomial Ring, Electronic dissertation published by ProQuest}, url={http://surveyor.gelman.gwu.edu/.}, author={Sazdanović, R.}, year={2010} } @article{jablan_radović_sazdanović_2010, title={Tutte and Jones polynomials of links, polyominoes and graphical recombination patterns}, volume={49}, ISSN={0259-9791 1572-8897}, url={http://dx.doi.org/10.1007/S10910-010-9731-1}, DOI={10.1007/S10910-010-9731-1}, number={1}, journal={Journal of Mathematical Chemistry}, publisher={Springer Science and Business Media LLC}, author={Jablan, S. and Radović, Lj. and Sazdanović, R.}, year={2010}, month={Oct}, pages={79–94} } @article{jablan_sazdanović_2008, title={BRAID FAMILY REPRESENTATIVES}, volume={17}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-49449083565&partnerID=MN8TOARS}, DOI={10.1142/s0218216508006427}, abstractNote={Imposing different conditions on minimality of reduced braid words and new criteria on their representatives, we define braid family representatives and establish one-to-one correspondence with families of knots and links given in Conway notation.}, number={07}, journal={Journal of Knot Theory and Its Ramifications}, publisher={World Scientific Pub Co Pte Lt}, author={JABLAN, SLAVIK and SAZDANOVIĆ, RADMILA}, year={2008}, month={Jul}, pages={817–833} } @article{pabiniak_przytycki_sazdanović_2008, title={On the first group of the chromatic cohomology of graphs}, volume={140}, ISSN={0046-5755 1572-9168}, url={http://dx.doi.org/10.1007/S10711-008-9307-4}, DOI={10.1007/S10711-008-9307-4}, abstractNote={The Hochschild homology of the algebra of truncated polynomials $${{\mathcal {A}_m=\mathbb {Z}[x]/(x^m)}}$$ is closely related to the Khovanov-type homology as shown by the second author. In the present paper we utilize this in the study of the first graph cohomology group of an arbitrary graph G with v vertices. The complete description of this group is given for m = 2, 3. For the algebra $${\mathcal {A}_2}$$ we relate the chromatic graph cohomology with the Khovanov homology of adequate links. We describe the chromatic cohomology over the algebra $${\mathcal {A}_3}$$ using the homology of a cell complex built on the graph G. In particular we prove that $${{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}$$ can be isomorphic to any finite abelian group. Moreover, we give a characterization of graphs which have torsion in cohomology $${{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}$$ and construct graphs which have the same (di)chromatic polynomial but different $${{H_{\mathcal {A}_{3}}^{1,2v-3}(G)}}$$ .}, number={1}, journal={Geometriae Dedicata}, publisher={Springer Science and Business Media LLC}, author={Pabiniak, Milena D. and Przytycki, Józef H. and Sazdanović, Radmila}, year={2008}, month={Nov}, pages={19–48} } @article{jablan_sazdanović_2007, title={Knots, Links, and Self-avoiding Curves}, volume={22}, number={1}, journal={Forma}, author={Jablan, S. and Sazdanović, R.}, year={2007}, pages={5–13} } @book{jablan_sazdanović_2007, title={Linknot - Knot Theory by Computer}, DOI={10.1142/9789812772244}, abstractNote={Basic Graph Theory Shadows of KLs Notation of Knots and Links (KLs) Gauss and Dowker Code KL Diagrams Reidemeister Moves Conway Notation Classification of KLs Chirality of KLs Unlinking Number and Unlinking Gap KLs with Unlinking Number One Non-Invertible KLs Braids Braid Family Representatives Borromean Links Recognition and Generation of KLs Polynomial Invariants Experimenting with KLs Derivation and Classification of KLs Basic Polyhedra, Polyhedral KLs, and Non-Algebraic Tangles Non-Alternating and Almost Alternating KLs Families of Undetectable KLs Detecting Chirality of KLs by Polynomial Invariants History of Knot Theory and Its Applications Mirror Curves KLs and Fullerenes KLs and Mathematical Logic Self-Referential Systems KL Automata.}, journal={Series on Knots and Everything}, publisher={World Scientific Publishing Co. Pte. Ltd.}, author={Jablan, Slavik and Sazdanović, Radmila}, year={2007} } @article{jablan_sazdanović_2007, title={UNLINKING NUMBER AND UNLINKING GAP}, volume={16}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-38549182277&partnerID=MN8TOARS}, DOI={10.1142/s0218216507005828}, abstractNote={Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard–Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking number. We compute BJ-unlinking number for various families of knots and links for which the unlinking number is unknown. Furthermore, we define BJ-unlinking gap and construct examples of links with arbitrarily large BJ-unlinking gap. Experimental results for BJ-unlinking gap of rational links up to 16 crossings, and all alternating links up to 12 crossings are obtained using programs LinKnot and K2K. Moreover, we propose families of rational links with arbitrarily large BJ-unlinking gap and polyhedral links with constant non-trivial BJ-unlinking gap. Computational results suggest existence of families of non-alternating links with arbitrarily large BJ-unlinking gap.}, number={10}, journal={Journal of Knot Theory and Its Ramifications}, publisher={World Scientific Pub Co Pte Lt}, author={JABLAN, SLAVIK and SAZDANOVIĆ, RADMILA}, year={2007}, pages={1331–1355} } @article{basic polyhedra in knot theory_2005, volume={28 }, journal={Kragujevac Journal of Mathematics}, year={2005}, pages={155–164} } @article{jablan_sazdanovic_2004, title={Discovering symmetry of knots by using program LinKnot}, volume={1-4}, journal={The Journal of ISIS-Symmetry}, author={Jablan, S. and Sazdanovic, R.}, year={2004}, pages={102–106} } @article{kappraff_jablan_adamson_sazdanović_2004, title={Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices}, volume={19}, number={4}, journal={Forma}, author={Kappraff, J. and Jablan, S. and Adamson, G.W. and Sazdanović, R.}, year={2004}, pages={367–387} } @article{sazdanovic_sremcević_2004, title={Hyperbolic Tessellations by tess}, journal={Symmetry: Art and Science (The Quarterly of ISIS Symmetry)}, author={Sazdanovic, R. and Sremcević, M.}, year={2004}, pages={1–4 226–229} } @inproceedings{sarhangi_jablan_sazdanovic_2004, title={Modularity in Medieval Persian Mosaics: Textual, Empirical, Analytical and Theoretical Considerations, Bridges}, booktitle={Mathematical Connections in Art, Music and Science, Conference Proceedings}, author={Sarhangi, R. and Jablan, S. and Sazdanovic, R.}, year={2004}, pages={281–293} } @article{jablan_sazdanovic_2004, title={Visualizing symmetry of knots by using program LinKnot, Symmetry: Art and Science}, volume={1-4}, journal={The Journal of ISIS-Symmetry}, author={Jablan, S. and Sazdanovic, R.}, year={2004}, pages={106–110} } @article{barrallo_sazdanovic_2002, title={Computer Sculpture: A Journey Through Mathematics,Bridges: Mathematical Connections in Art}, volume={54}, journal={Music and Science, Conference Proceedings}, author={Barrallo, J. and Sazdanovic, R.}, year={2002} } @article{sazdanovic_sremcevic_2002, title={Tessellations of the Euclidean, Elliptic and Hyperbolic Plane, Symmetry}, volume={2}, journal={Art and Science}, author={Sazdanovic, R. and Sremcevic, M.}, year={2002}, pages={229–304} } @article{jablan_radovic_sazdanovic, place={PO BOX 60, RADOJA DOMANOVICA 12, KRAGUJEVAC 34000, SERBIA}, title={BASIC POLYHEDRA IN KNOT THEORY}, volume={28}, journal={KRAGUJEVAC JOURNAL OF MATHEMATICS}, publisher={UNIV KRAGUJEVAC, FAC SCIENCE}, author={Jablan, Slavik V. and Radovic, Ljiljana M. and Sazdanovic, Radmila}, pages={{155–164}} } @book{levitt_hajij_sazdanovic, title={Big Data approaches to knot theory: Understanding the structure of the Jones polynomial}, number={1912.10086}, author={Levitt, J.S. and Hajij, M. and Sazdanovic, R.} }