Cocycle Knot Invariants, Quandle Extensions, and Alexander Matrices

TL;DR

This paper reviews recent developments in quandle cohomology and cocycle knot invariants, introduces dynamical cocycles for coloring knotted surfaces, and explores their connections to Alexander matrices.

Contribution

It introduces the concept of dynamical cocycles and demonstrates their application to coloring knotted surfaces and relating cocycle invariants to Alexander matrices.

Findings

Dynamical cocycles can be used to color knotted surfaces from classical knots.

Relations between cocycle invariants and Alexander matrices are established.

Summary of recent advances in quandle (co)homology and cocycle knot invariants.

Abstract

The theory of quandle (co)homology and cocycle knot invariants is rapidly being developed. We begin with a summary of these recent advances. One such advance is the notion of a dynamical cocycle. We show how dynamical cocycles can be used to color knotted surfaces that are obtained from classical knots by twist-spinning. We also demonstrate relations between cocycle invariants and Alexander matrices.