Research On Khovanov Type Homology Theory

Khovanov homology is a categorification of Jones polynomial. Since M. Khovanov built up Khovanov theory in 1999, it has been developed by many topologists. In recent years, there has been tremendous interest in developing Khovanov homology theory. Currently, the generalized Khovanov theories and their corresponding computations are the hot issues.This paper constructs a new " Khovanov type " homology theory, which is a natural gener-alization to the original Khovanov homology theory. We give the detailed geometric descriptions for this new homology theory and calculate the Khovanov homology of Kanenobu knots and the Khovanov type homology of pretzel knots, respectively. The main contents are as follows:1. We give a recursive formula for the Khovanov homology of all Kanenobu knots K(p, q) over F2. The result implies that the rank of the Khovanov homology of K{p, q) is a function of p+q. Our computation uses only the basic long exact sequence and some results on Khovanov-thin knots.2. We construct a new Khovanov type homology from a Frobenius algebra by applying a topology quantum field theory (TQFT). We show that the homology is an invariant for ori-ented links. We also get geometric descriptions by introducing the genus generating operations. We prove that the homology is a categorification of Jones Polynomial. As an application, we compute the homology groups of (2,k)-torus knots for every k?N.3. We give a recursive formula for Khovanov type homology of pretzel knots P(-n, -m, m) over R. The computations reveal that the rank of Khovanov type homology of the pretzel knots is an invariant of n. The proof is based on two resolutions of a crossing that reduce the compu-tational complexity of Khovanov type homology, thus we obtain a new method to calculate the link homology of the knots.