Twisted quandle homology theory and cocycle knot invariants

Abstract

The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite 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. Explicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The corresponding generalization of the quandle cocycle knot invariants is given, by using the Alexander numbering of regions in the definition of state-sums. The invariants are used to derive information on twisted cohomology groups.

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-6.abs.html

Cited by in corpus (26)

• Automorphism groups of Quandles
• Link invariants from finite racks
• Distributivity in Quandles and Quasigroups
• Cocycle Knot Invariants from Quandle Modules and Generalized Quandle Cohomology
• Heap and Ternary Self-Distributive Cohomology
• Quandle cohomology, extensions and automorphisms
• Higher Arity Self-Distributive Operations in Cascades and their Cohomology
• Quandle homology and complex volume
• Cocycle Knot Invariants, Quandle Extensions, and Alexander Matrices
• Positive quandle homology and its applications in knot theory
• Cohomology of Categorical Self-Distributivity
• RNA foldings, Oriented Stuck Knots and State Sum Invariants
• Continuous cohomology of topological quandles
• The shadow nature of positive and twisted quandle cocycle invariants of knots
• $f$-Racks, $f$-Quandles, their Extensions and Cohomology
• Entropic magmas, their homology, and related invariants of links and graphs
• N-Degeneracy in rack homology and link invariants
• Cohomology of Frobenius Algebras and the Yang-Baxter Equation
• A Diagrammatic Construction of Third Homology Classes of Knot Quandles
• Polynomial cocycles of Alexander quandles and applications
• Virtual Yang-Baxter cocycle invariants
• Three geometric applications of quandle homology
• Ternary and $n$-ary $f$-distributive Structures
• Integral positive (negative) quandle cocycle invariants are trivial for knots
• Cocycle Deformations of Algebraic Identities and R-matrices
• From racks to pointed Hopf algebras