Three new knot invariants are defined using cocycles
of the generalized quandle homology theory that was proposed by
Andruskiewitsch and Graña.
We specialize that theory to the case when there is a group action on
the coefficients.
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First, quandle modules are used to generalize Burau representations
and Alexander modules for classical knots. Second, 2-cocycles
valued in non-abelian groups are used in a way similar to Hopf algebra
invariants of classical knots.
These invariants are shown to be of quantum type.
Third, cocycles with group actions on
coefficient groups are used to define
quandle cocycle
invariants for both
classical knots and knotted surfaces. Concrete computational methods
are provided
and used to prove non-invertibility for a large family of knotted surfaces.
In the classical case, the invariant
can detect the chirality
of 3-colorable knots in a number of cases.