## 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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## Additional Information

**Yu Hashimoto**- Affiliation: Toshimagaoka-jyoshigakuen High School, 1-25-22, Higashi-ikebukuro, Toshima-ku, Tokyo 170-0013, Japan
**Kokoro Tanaka**- Affiliation: Department of Mathematics, Tokyo Gakugei University, Nukuikita 4-1-1, Koganei, Tokyo 184-8501, Japan
- MR Author ID: 752455
- ORCID: 0000-0003-0815-0081
- Email: kotanaka@u-gakugei.ac.jp
- Received by editor(s): August 12, 2021
- Received by editor(s) in revised form: March 6, 2022
- Published electronically: July 29, 2022
- Additional Notes: The second author has been supported in part by the Grant-in-Aid for Scientific Research (C), (No. JP17K05242, No. JP21K03220) Japan Society for the Promotion of Science.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 7261-7276 - MSC (2020): Primary 57K12, 57K10; Secondary 57K45, 55N99
- DOI: https://doi.org/10.1090/tran/8707