Displaying similar documents to “Matrix factorizations and link homology”

A colored 𝔰𝔩(N) homology for links in S³

Hao Wu

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Fix an integer N ≥ 2. To each diagram of a link colored by 1,...,N we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain complex defined by Khovanov and Rozansky. The homology of this chain complex decategorifies to the Reshetikhin-Turaev 𝔰𝔩(N) polynomial of links colored by exterior powers...

Khovanov homology, its definitions and ramifications

Oleg Viro (2004)

Fundamenta Mathematicae

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Mikhail Khovanov defined, for a diagram of an oriented classical link, a collection of groups labelled by pairs of integers. These groups were constructed as the homology groups of certain chain complexes. The Euler characteristics of these complexes are the coefficients of the Jones polynomial of the link. The original construction is overloaded with algebraic details. Most of the specialists use adaptations of it stripped off the details. The goal of this paper is to overview these...

Chewing the Khovanov homology of tangles

Magnus Jacobsson (2004)

Fundamenta Mathematicae

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We present an elementary description of Khovanov's homology of tangles [K2], in the spirit of Viro's paper [V]. The formulation here is over the polynomial ring ℤ[c], unlike [K2] where the theory was presented over the integers only.

A colored Khovanov bicomplex

Noboru Ito (2014)

Banach Center Publications

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In this note, we prove the existence of a tri-graded Khovanov-type bicomplex (Theorem 1.2). The graded Euler characteristic of the total complex associated with this bicomplex is the colored Jones polynomial of a link. The first grading of the bicomplex is a homological one derived from cabling of the link (i.e., replacing a strand of the link by several parallel strands); the second grading is related to the homological grading of ordinary Khovanov homology; finally, the third grading...

New categorifications of the chromatic and dichromatic polynomials for graphs

Marko Stošić (2006)

Fundamenta Mathematicae

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For each graph G, we define 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 define 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 categorifications of the HOMFLYPT polynomial. We also give a simplified definition...

A 2-category of chronological cobordisms and odd Khovanov homology

Krzysztof K. Putyra (2014)

Banach Center Publications

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We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms along any 3-dimensional subcube is trivial. This allows us to create a chain complex whose homotopy type modulo certain relations is a link invariant. Both the original and...

Torsion of Khovanov homology

Alexander N. Shumakovitch (2014)

Fundamenta Mathematicae

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Khovanov homology is a recently introduced invariant of oriented links in ℝ³. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of Khovanov homology is a version of the Jones polynomial for links. In this paper we study torsion of Khovanov homology. Based on our calculations, we formulate several conjectures about the torsion and prove weaker versions of the first two of them. In particular, we prove that all non-split alternating links have their...

Introduction to the basics of Heegaard Floer homology

Bijan Sahamie (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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This paper provides an introduction to the basics of Heegaard Floer homology with some emphasis on the hat theory and to the contact geometric invariants in the theory. The exposition is designed to be comprehensible to people without any prior knowledge of the subject.

Homology of representable sets

Marian Mrozek, Bogdan Batko (2010)

Annales Polonici Mathematici

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We generalize the notion of cubical homology to the class of locally compact representable sets in order to propose a new convenient method of reducing the complexity of a set while computing its homology.

Khovanov-Rozansky homology for embedded graphs

Emmanuel Wagner (2011)

Fundamenta Mathematicae

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For any positive integer n, Khovanov and Rozansky constructed a bigraded link homology from which you can recover the 𝔰𝔩ₙ link polynomial invariants. We generalize the Khovanov-Rozansky construction in the case of finite 4-valent graphs embedded in a ball B³ ⊂ ℝ³. More precisely, we prove that the homology associated to a diagram of a 4-valent graph embedded in B³ ⊂ ℝ³ is invariant under the graph moves introduced by Kauffman.

A computation in Khovanov-Rozansky homology

Daniel Krasner (2009)

Fundamenta Mathematicae

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We investigate the Khovanov-Rozansky invariant of a certain tangle and its compositions. Surprisingly the complexes we encounter reduce to ones that are very simple. Furthermore, we discuss a "local" algorithm for computing Khovanov-Rozansky homology and compare our results with those for the "foam" version of sl₃-homology.

Transverse Homology Groups

S. Dragotti, G. Magro, L. Parlato (2006)

Bollettino dell'Unione Matematica Italiana

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We give, here, a geometric treatment of intersection homology theory.

On the geometry of polynomial mappings at infinity

Anna Valette, Guillaume Valette (2014)

Annales de l’institut Fourier

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We associate to a given polynomial map from ℂ 2 to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

Steenrod homology

Yu. T. Lisitsa, S. Mardešić (1986)

Banach Center Publications

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