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Braid Family RepresentativesApr 23 2005After defining reduced minimum braid word and criteria for a braid family representative, different braid family representatives are derived, and a correspondence between them and families of knots and links given in Conway notation is established.

Patterns in Khovanov link and chromatic graph homologyJan 04 2018Khovanov 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 ... More

Categorification of the polynomial ringDec 31 2010We develop a diagrammatic categorification of the polynomial ring $Z[x]$. Our categorification satisfies a version of Bernstein-Gelfand-Gelfand reciprocity property with the indecomposable projective modules corresponding to $x^n$ and standard modules ... More

Graph homology and graph configuration spacesAug 28 2012If $R$ 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, ... More

Chromatic homology, Khovanov homology, and torsionSep 12 2016In 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 the categorification of the chromatic ... More

Torsion in Khovanov homology of semi-adequate linksOct 18 2012The goal of this paper is to address A. Shumakovitch's conjecture about the existence of $\Z_2$-torsion in Khovanov link homology. We analyze torsion in Khovanov homology of semi-adequate links via chromatic cohomology for graphs which provides a link ... More

Chromatic homology, Khovanov homology, and torsionSep 12 2016Mar 14 2017In 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 the categorification of the chromatic ... More

Tutte and Jones polynomials of link familiesApr 24 2010This article contains general formulas for Tutte and Jones polynomials for families of knots and links given in Conway notation and "portraits of families"-- plots of zeroes of their corresponding Jones polynomials.

Psyquandles, Singular Knots and PseudoknotsOct 23 2017We 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 ... More

On the first group of the chromatic cohomology of graphsJul 13 2006The algebra of truncated polynomials A_m=Z[x]/(x^m) plays an important role in the theory of Khovanov and Khovanov-Rozansky homology of links. We have demonstrated that Hochschild homology is closely related to Khovanov homology via comultiplication free ... More

Reduced Relative Tutte, Kauffman Bracket and Jones Polynomials of Virtual Link FamiliesJun 14 2011Feb 05 2013This 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 ... More

Mirror-Curves and Knot MosaicsJun 19 2011Dec 28 2011Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid ... More

Torsion in thin regions of Khovanov homologyMar 13 2019In 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 only contain $\mathbb{Z}_2$ torsion. In this paper, we ... More

A categorification of the chromatic symmetric functionJun 09 2015The Stanley chromatic symmetric function $X_G$ of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology of graded $S_n$-modules, ... More

Vietoris-Rips and Cech Complexes of Metric GluingsDec 18 2017May 27 2018We 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. ... More

Tutte and Jones Polynomial of Link FamiliesFeb 06 2009Apr 24 2010This article contains general formulas for Tutte and Jones polynomial for families of knots and links given in Conway notation.

Unlinking Number and Unlinking GapMar 14 2005Nov 21 2007Computing 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 ... More

The Relationship Between the Intrinsic Cech and Persistence Distortion Distances for Metric GraphsDec 13 2018Metric 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 ... More

A Complete Characterization of the 1-Dimensional Intrinsic Cech Persistence Diagrams for Metric GraphsFeb 23 2017Jul 07 2017Metric 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 ... More

Quasi-alternating links and odd homology: computations and conjecturesDec 31 2008Mar 29 2014We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating ... More

Comparing Multilayer Perceptron and Multiple Regression Models for Predicting Energy Use in the BalkansOct 26 2018Global demographic and economic changes have a critical impact on the total energy consumption, which is why demographic and economic parameters have to be taken into account when making predictions about the energy consumption. This research is based ... More

Multi-target Radar Detection within a Sparsity FrameworkNov 06 2013Traditional radar detection schemes are typically studied for single target scenarios and they can be non-optimal when there are multiple targets in the scene. In this paper, we develop a framework to discuss multi-target detection schemes with sparse ... More