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An introduction to Khovanov homology categorification of skein modules
by
Jozef H. Przytycki
George Washington University
Coauthors: Marta M. Asaeda (U.Iowa), Adam S. Sikora (SUNY, Buffalo)

Khovanov homology offers a nontrivial generalization of the Jones polynomials of links in R³ (and of the Kauffman bracket skein modules of some 3-manifolds). We define Khovanov homology of links in R³ 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 long exact sequence of Khovanov 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. We define Khovanov homology, H_{i, j, k} for links in products of surfaces and an interval and in twisted I-bundles over unorientable surfaces (excluding RP²). We show how to stratify this homology so that (in the product case, F×I) it categorifies the Kauffman bracket skein module (KBSM) of F×I. That is, for any link L in F×I we can recover coefficients of L in the standard basis B(F) of the KBSM of F×I. In other words if L = Sum_b a_b(A) b where the sum is taken over all basic elements, b in B(F), then each coefficient a_b(A) can be recovered from polynomial Euler characteristics of the stratified Khovanov homology. In the case of unorientable F we are able to recover coefficients a_b(A) only partially. We propose another basis of the KBSM for which categorification seems to be possible even for unorientable F (we use cores of Möbius bands several times even if they intersect one another).

Date received: November 13, 2004

Copyright © 2004 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # capf-13.