Lipshitz-Sarkar refines mirrors more than Khovanov

Yang-Hui He

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arXiv · 2609.00976

Lipshitz-Sarkar refines mirrors more than Khovanov

Yang-Hui He

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Abstract

Lipshitz and Sarkar asked in an ICM 2018 question whether their homotopy-theoretic refinement of Khovanov homology can detect mirror reflection when ordinary integral Khovanov homology cannot. We give an affirmative answer by presenting a prime knot whose integral Khovanov homology agrees with that of its mirror in every bidegree, including torsion, and yet over $\mathbb{F}_2$, the second Steenrod square has rank one for the knot and rank zero for its mirror in a specified bidegree. We also give a composite example. The construction, search and computations are done with the help of {\it chat-GTP 5.6 sol Pro} combing over the {\it Regina Census} of knots.

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Yang-Hui He. 2026-09-01. Lipshitz-Sarkar refines mirrors more than Khovanov. https://arxiv.org/abs/2609.00976

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