Khovanov Homology: categorification of the Kauffman bracket relation

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dc.creatorPrzytycki, Jozef H.
dc.date2005-12-29
dc.date.accessioned2026-07-07T06:55:51Z
dc.date.available2026-07-07T06:55:51Z
dc.descriptionKhovanov homology offers a nontrivial generalization of Jones polynomial of links in R^3 (and of Kauffman bracket skein module of some 3-manifolds). In this chapter (Chapter X) we define Khovanov homology of links in R^3 and generalize the construction into links in an I-bundle over a surface. We use Viro's approach to construction of Khovanov homology and utilize the fact that one works with unoriented diagrams (unoriented framed links) in which case there is a simple description of Khovanov long exact sequence of homology. Khovanov homology, over the field Q, is a categorification of the Jones polynomial (i.e. one represents the Jones polynomial as the generating function of Euler characteristics). However, for integral coefficients Khovanov homology almost always has torsion. The first part of the chapter is devoted to the construction of torsion in Khovanov homology. In the second part we analyze the thickness of Khovanov homology and reduced Khovanov homology. In the last part we generalize Khovanov homology to links in $I$-bundles over surfaces and demonstrate that for oriented surfaces we categorify the element of the Kauffman bracket skein module represented by the considered link.
dc.description79 pages, 73 figures, Chapter X of the Book "KNOTS: From combinatorics of knot diagrams to combinatorial topology based on knots
dc.identifierhttps://arxiv.org/abs/math/0512630
dc.identifierhttp://arxiv.org/abs/math/0512630
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/106421
dc.subjectGeometric Topology
dc.subjectAlgebraic Topology
dc.subject57M (primary), 14F (secondary)
dc.titleKhovanov Homology: categorification of the Kauffman bracket relation
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