Twisted quandle homology theory and cocycle knot invariants

By J. Scott Carter, Mohamed Elhamdadi, Masahico Saito. 2001

70 citations recorded by the source corpus.

The quandle homology theory is generalized to the case when the coecient groups admit the structure of Alexander quandles, by includ- ing an action of the innite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Ex- plicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The cor- responding generalization of the quandle cocycle knot invariants is given, by using the Alexander numbering of regions in the denition of state- sums. The invariants are used to derive information on twisted cohomology groups. AMS Classication 57N27, 57N99; 57M25, 57Q45, 57T99

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d212db80d8ca0019768562a2544d3819c5ed1def

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Citations in the corpus

Cohomology of Categorical Self-Distributivity (2006, unknown): “Generalizations have been discovered [1, 8, 9].”
Cocycle knot invariants from quandle modules and generalized quandle homology (2003, background): “The invariant agrees with the quandle cocycle invariant Φφ(K) defined in [10] when the quandle module structure of the coefficient group G is trivial, and with Φφ(K) defined in [7], modulo the Alexander numbering convention in the Boltzmann weight in [7], when the coefficient group G is a Z[t, t−1]-”
Extensions of Quandles and Cocycle Knot Invariants (2001, background): “An extension theory of quandles for twisted cohomology cocycles was developed in [6], and it provided more general extension theories.”
Homologies of Algebraic Structures via Braidings and Quantum Shuffles (2012, background): “The differential d− dT defines the twisted rack homology from [3].”