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PreprintCC BY-NC-ND 4.0arXiv

The flip symmetry on Khovanov-Rozansky homology

  • Hongjian Yang
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Abstract

The flip symmetry on link diagrams induces an involution on Khovanov-Rozansky frakglN homology. We prove that this involution is diagonalizable with eigenvalues ±1. On the one hand, it is the identity over BbbF2, generalizing a previous result of Chen and the author. On the other hand, it is expected to be nontrivial over BbbZ in general. The key ingredients of the proof are (1) a homotopy perturbation argument via a detailed study of the fork twist, which allows us to reduce the computation to planar frakglN webs, and (2) the computation of the flip map for planar frakglN webs via diagrammatics of Soergel bimodules. The latter computation can also be interpreted as a naturality result for the half twist action on type A Soergel bimodules, which might be of independent interest.

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Cite

BibTeXRIS
BibTeX
@misc{yang2026flip,
  title = {{The flip symmetry on Khovanov-Rozansky homology}},
  author = {Hongjian Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18780},
  eprint = {2609.18780},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18780},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - The flip symmetry on Khovanov-Rozansky homology
AU  - Hongjian Yang
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18780
UR  - https://arxiv.org/abs/2609.18780
AB  - The flip symmetry on link diagrams induces an involution on Khovanov-Rozansky frakgl_N homology. We prove that this involution is diagonalizable with eigenvalues ±1. On the one hand, it is the identity over BbbF_2, generalizing a previous result of Chen and the author. On the other hand, it is expected to be nontrivial over BbbZ in general. The key ingredients of the proof are (1) a homotopy perturbation argument via a detailed study of the fork twist, which allows us to reduce the computation to planar frakgl_N webs, and (2) the computation of the flip map for planar frakgl_N webs via diagrammatics of Soergel bimodules. The latter computation can also be interpreted as a naturality result for the half twist action on type A Soergel bimodules, which might be of independent interest.
ER  -