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arXiv · 2609.18780

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 ${\frak{gl}}_{N}$ homology. We prove that this involution is diagonalizable with eigenvalues $\pm1$. On the one hand, it is the identity over $\Bbb{F}_2$, generalizing a previous result of Chen and the author. On the other hand, it is expected to be nontrivial over $\Bbb{Z}$ 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 ${\frak{gl}}_{N}$ webs, and (2) the computation of the flip map for planar ${\frak{gl}}_{N}$ 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.

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Hongjian Yang. 2026-09-16. The flip symmetry on Khovanov-Rozansky homology. https://arxiv.org/abs/2609.18780

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