Algebraic and Computational Aspects of Quandle 2-Cocycle Invariant

Abstract

Quandle homology theories have been developed and cocycles have been used to construct invariants in state-sum form for knots using colorings of knot diagrams by quandles. Quandle 2-cocycles can be also used to define extensions as in the case of groups. There are relations among algebraic properties of quandles, their homology theories, and cocycle invariants; certain algebraic properties of quandles affect the values of the cocycle invariants, and identities satisfied by quandles induce subcomplexes of homology theory. Recent developments in these matters, as well as computational aspects of the invariant, are reviewed. Problems and conjectures pertinent to the subject are also listed.