Shifting chain maps in quandle homology and cocycle invariants

Hashimoto, Yu; Tanaka, Kokoro

doi:10.1090/tran/8707

Shifting chain maps in quandle homology and cocycle invariants
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by Yu Hashimoto and Kokoro Tanaka PDF

Trans. Amer. Math. Soc. 375 (2022), 7261-7276 Request permission

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 $\sigma ^\sharp$, each $2$-cocycle $\phi$ gives us the $3$-cocycle $\sigma ^\sharp \phi$. 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 $\sigma ^\sharp \phi$. For oriented surface links in the $4$-space, we explore how powerful their quandle $3$-cocycle invariants associated with $\sigma ^\sharp \phi$ are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.

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