Cocycle knot invariants from quandle modules and generalized quandle homology

Abstract

Three new knot invariants are defined using cocycles\nof the generalized quandle homology theory that was proposed by\nAndruskiewitsch and Graña.\nWe specialize that theory to the case when there is a group action on\nthe coefficients.\n¶\nFirst, quandle modules are used to generalize Burau representations\nand Alexander modules for classical knots. Second, 2-cocycles\nvalued in non-abelian groups are used in a way similar to Hopf algebra\ninvariants of classical knots.\nThese invariants are shown to be of quantum type.\nThird, cocycles with group actions on\ncoefficient groups are used to define\nquandle cocycle\ninvariants for both\nclassical knots and knotted surfaces. Concrete computational methods\nare provided\nand used to prove non-invertibility for a large family of knotted surfaces.\nIn the classical case, the invariant\ncan detect the chirality\nof 3-colorable knots in a number of cases.