arXiv subjects

Marko Stosic

Publications and source records attributed to Marko Stosic.

24 records · Page 2Linked to original sources

arXiv2006

Homological thickness and stability of torus knots

In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, we show that there exists stable homology of torus knots conjectured by Dunfield, Gukov and Rasmussen in \cite{dgr}. Since our main tool is the long exact sequence in homology, we have applied our approach in the case of the Khovanov-Rozansky ($sl(n)$) homology, and thus obtained analogous stability properties of $sl(n)$ homology of torus knots, also conjectured in \cite{dgr}.

math.GT↗

arXiv2006

New Categorifications of the Chromatic and the Dichromatic Polynomials for Graphs

In this paper, for each graph $G$, we def\mbox{}ine a chain complex of graded modules over the ring of polynomials, whose graded Euler characteristic is equal to the chromatic polynomial of $G$. Furthermore, we def\mbox{}ine a chain complex of doubly-graded modules, whose (doubly) graded Euler characteristic is equal to the dichromatic polynomial of $G$. Both constructions use Koszul complexes, and are similar to the new Khovanov-Rozansky categorif\mbox{}ications of HOMFLYPT polynomial. We also give simplif\mbox{}ied def\mbox{}inition of this triply-graded link homology theory.

math.QA↗

arXiv2006

Khovanov homology of links and graphs

In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are link invariants, up to chain homotopy, with graded Euler characteristic equal to the Jones polynomial of the link. Hence, it can be regarded as the "categorif\mbox{}ication" of the Jones polynomial. \indent We prove that the f\mbox{}irst homology group of positive braid knots is trivial. Futhermore, we prove that non-alternating torus knots are homologically thick. In addition, we show that we can decrease the number of full twists of torus knots without changing low-degree homology and consequently that there exists stable homology for torus knots. We also prove most of the above properties for Khovanov-Rozansky homology. \indent Concerning graph homology, we categorify the dichromatic (and consequently Tutte) polynomial for graphs, by categorifying an inf\mbox{}inite set of its one-variable specializations. We categorify explicitly the one-variable specialization that is an analog of the Jones polynomial of an alternating link corresponding to the initial graph. Also, we categorify explicitly the whole two-variable dichromatic polynomial of graphs by using Koszul complexes. \textbf{Key-words:} Khovanov homology, Jones polynomial, link, torus knot, graph, dichromatic polynomial

math.QA↗

arXiv2005

Properties of Khovanov homology for positive braid knots

In this paper we solve one open problem from \cite{pat} and give some generalizations. Namely, we prove that the first homology group of positive braid knot is trivial. Also, we show that the same is true for the Khovanov-Rozansky homology \cite{kovroz} ($sl(n)$ link homology) for any positive integer $n$.

math.QA↗

arXiv2005

Categorification of the Dichromatic Polynomial for Graphs

For each graph and each positive integer $n$, we define a chain complex whose graded Euler characteristic is equal to an appropriate $n$-specialization of the dichromatic polynomial. This also gives a categorification of $n$-specializations of the Tutte polynomial of graphs. Also, for each graph and integer $n\le 2$, we define the different one variable $n$-specializations of the dichromatic polynomials, and for each polynomial we define graded chain complex whose graded Euler characteristic is equal to that polynomial. Furthermore, we explicitly categorify the specialization of the Tutte polynomial for graphs which corresponds to the Jones polynomial of the appropriate alternating link.

math.GT↗

arXiv2004

Parasupersymmetric Quantum Mechanics of Order 3 and a Generalized Witten Index

In this paper we generalize the Rubakov-Spiridonov parasupersymmetry algebra to the order 3 case. We also generalize the notion of the Witten index, and we provide a class of models satisfying our parasupersymmetry algebra. Finally, we show that there is a correspondence between the Hamiltonian and the index in our class of models.

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