Abstract:

Khovanov homology is a categorification of the Jones polynomial of links. As written in Viro's paper (O.Viro, Fund. Math. 184 (2004), 317--342), "the most fundamental property of the Khovanov homology groups is their invariance under Reidemeister moves". Khovanov homology groups are knot invariants because these groups are invariant under three types of Reidemeister moves. By giving explicit chain homotopy maps using Viro's definition of the homology, he proved the invariance under the first Reidemeister moves. This talk gives chain homotopy maps ensuring the invariance under the other Reidemeister moves. We discuss a good property of the explicit chain homotopy maps. If time permits, we also discuss the existence of a bicomplex which is a Khovanov-type complex associated with categorification of colored Jones polynomial. This is an answer to the question proposed by A. Beliakova and S. Wehrli. Then the second term of the spectral sequence of the bicomplex corresponds to the Khovanov-type homology group.