Cocycle knot invariants from quandle modules and generalized quandle homology

By J. Scott Carter, Mohamed Elhamdadi, M. Graña, Masahico Saito. 2003

108 citations recorded by the source corpus.

Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Grana. We specialize that theory to the case when there is a group action on the coefficients. 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b7b28f0b38349688a3c39585dbdc2343547082bd

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Homology Groups of Symmetric Quandles and Cocycle Invariants of Links and Surface-links (2009, background): “For details on quandle homologies and invariants, refer to [1, 3, 4, 5, 10, 11, 16, 20, 21].”
Matrices and Finite Quandles (2004, methodology): “quandle to a chosen ﬁnite quandle. One can also obtain knot invariants by counting homomorphisms with crossings weighted by quandle cocyles arising in various quandle cohomology theories. See [1] and [2] for more. In this paper, we show how to associateto any ﬁnite quandle Q = {x1,x2,...,xn} an n × ”