Shifting chain maps in quandle homology and cocycle invariants (2012.09584v2)

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.