Generalizations of Quandle Cocycle Invariants and Alexander Modules from Quandle Modules

Quandle cohomology theory was developed [5] to define invariants of classical knots and knotted surfaces in state-sum form, called quandle cocycle (knot) invariants. The quandle cohomology theory is a modification of rack cohomology theory which was defined in [11]. The cocycle knot invariants are analogous in their definitions to the Dijkgraaf-Witten invariants [8] of triangulated 3-manifolds with finite gauge groups, but they use quandle knot colorings as spins and cocycles as Boltzmann weights. In [4], the quandle cocycle invariants were generalized in three different directions, using generalizations of quandle homology theory provided by Andruskiewitsch and Graña [1], which is compared to the group cohomology theories with the group actions on the coefficient groups. This paper is a written version of our talk given at Intelligence of Low Dimensional Topology in Shodo-Shima. It is a short summary of [4] with some results from [6] and a few new observations. We would like to thank the organizers for holding such an exciting conference in a beautiful location.