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Computing the Khovanov Homology with KhoHo
by
Alexander Shumakovitch
Dartmouth College
Given a diagram D of an oriented link L in the 3-sphere, one can assign to it a family of Abelian groups Hi, j(D) using a construction due to Mikhail Khovanov. These groups are defined as homology groups of an appropriate (graded) chain complex C(D), and their isomorphism classes depend on the isotopy class of L only. The graded Euler characteristic of C(D) is a version of the Jones polynomial of L.
One of the most remarkable properties of the Khovanov homology is that their ranks are much smaller than the ones of the original chain complex. For example, the chain groups of the torus knot (2, 11) have ranks as high as 11'000, but the homology themselves are at most Z. This property makes it utterly complicated to compute the Khovanov homology for knots and links with even 10 crossings.
In this talk, we show how one can dramatically reduce the size of the complex, without changing its homology, with the help of two elementary operations. This simplification procedure was implemented in the program KhoHo. We will see how to use KhoHo to compute and study Khovanov Homology.
Date received: December 10, 2003
Copyright © 2003 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 # camw-16.