PreprintCC BY 4.0arXiv
Lipshitz-Sarkar refines mirrors more than Khovanov
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 𝔽2, 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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BibTeX
@misc{he2026lipshitzsarkar,
title = {{Lipshitz-Sarkar refines mirrors more than Khovanov}},
author = {Yang-Hui He},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.00976},
eprint = {2609.00976},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00976},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lipshitz-Sarkar refines mirrors more than Khovanov
AU - Yang-Hui He
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2609.00976
UR - https://arxiv.org/abs/2609.00976
AB - 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 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.
ER -