Shifting chain maps in quandle homology and cocycle invariants

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 $\varphi$ gives us the $3$-cocycle ${\sigma }^{\#}\varphi$. For oriented classical links in the $3$-space, we explore relation between their quandle $2$-cocycle invariants associated with $\varphi$ and their shadow $3$-cocycle invariants associated with ${\sigma }^{\#}\varphi$. For oriented surface links in the $4$-space, we explore how powerful their quandle $3$-cocycle invariants associated with ${\sigma }^{\#}\varphi$ are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.

16 pages. Problem 6.7(v1) is replaced with Conjecture 6.7(v2). Problem 6.14(v1) is solved and hence deleted. As a result, Proposition 6.13(v1) is refined into Theorem 6.13(v2)