Shifting chain maps in quandle homology and cocycle invariants

TL;DR

This paper introduces a shifting chain map in quandle homology that relates 2- and 3-cocycle invariants, enhancing understanding of link invariants in classical and surface links.

Contribution

It defines a new shifting chain map in quandle homology and explores its impact on cocycle invariants for classical and surface links, revealing new relationships.

Findings

Relation between quandle 2-cocycle and shadow 3-cocycle invariants for classical links

Enhanced power of 3-cocycle invariants for surface links in 4-space

Analysis of algebraic behavior of shifting maps in low-dimensional (co)homology

Abstract

Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map $\sigma$ on each quandle chain complex that lowers the dimensions by one. By using its pull-back $\sigma^\#$, each $2$-cocycle $\phi$ gives us the $3$-cocycle $\sigma^\# \phi$. For oriented classical links in the $3$-space, we explore relation between their quandle $2$-cocycle invariants associated with $\phi$ and their shadow $3$-cocycle invariants associated with $\sigma^\# \phi$. For oriented surface links in the $4$-space, we explore how powerful their quandle $3$-cocycle invariants associated with $\sigma^\# \phi$ are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.