Khovanov Homology as an Generalization of the Jones Polynomial in Kauffman Terms

Author

Heather Tram, Southern Illinois University CarbondaleFollow

Date of Award

8-1-2016

Degree Name

Master of Science

Department

Mathematics

First Advisor

Kocik, Jerzy

Abstract

This paper explains the construction of Khovanov homology of which begins by un derstanding how Louis Kauffman generalizes the Jones polynomial using a state sum model of the bracket polynomial for an unoriented knot or link and in turn recovers the Jones polynomial, a knot invariant for an oriented knot or link. Kauffman associates the unknot by the polynomial (−A2 − A−2) whereas Khovanov associates the unknot by (q + q−1) through a change of variables. As an oriented knot or link K with n crossings produces 2n smoothings, Khovanov builds a commutative cube {0,1}n and associates a graded vector space to each smoothing in the cube. By defining a differential operator on the directed edges of the cube so that adjacent states differ by a type of smoothing for a fixed cross ing, we can form chain groups which are direct sums of these vector spaces. Naturally we get a bi-graded (co)chain complex which is called the Khovanov complex. The resulting (co)homology groups of these (co)chains turns out to be invariant under the Reidemeister moves and taking the Euler characteristic of the Khovanov complex returns the very same Jones polynomial that we started with.