Lipshitz-Sarkar refines mirrors more than Khovanov

Technical Summary: "Lipshitz-Sarkar Refines Mirrors More Than Khovanov"

Problem Statement
This paper addresses Question 2 (Q2) from a list of 11 problems posed by Lipshitz and Sarkar at the 2018 International Congress of Mathematicians (ICM). The question asks whether the homotopy-theoretic refinement of Khovanov homology (the Khovanov spectrum) provides a stronger obstruction to amphichirality (mirror symmetry) than ordinary integral Khovanov homology. Specifically, the author seeks to determine if there exists a knot $K$ and its mirror $mK$ such that their integral Khovanov homology groups are isomorphic in every bidegree (including torsion), yet their Khovanov spectra are not equivalent.

Methodology
The author employs a computer-assisted search strategy utilizing the Regina Census of knots, aided by the AI tool "chat-GTP 5.6 sol Pro." The methodology involves a multi-stage filtering process applied to the census of prime knots:

1. Rational Check: Knots are first filtered by comparing their rational Khovanov polynomials with those of their mirrors.
2. Integral Check: Candidates surviving the first stage are checked for isomorphism in complete integral Khovanov homology groups, including torsion subgroups.
3. Support Check: The remaining candidates are screened to ensure their mod-2 homology tables contain source and target groups separated by two cohomological degrees, a necessary condition for the Steenrod square operation ($Sq^2$) to be non-trivial.
4. Geometric Symmetry Check: An independent check using SnapPy verifies that the knot's complement has a trivial symmetry group, ensuring the knot is not inherently mirror-symmetric.
5. Spectral Computation: For the final candidates, the author computes the rank of the Steenrod square map $Sq^2: Kh_{i,j}(K; \mathbb{F}_2) \to Kh_{i+2,j}(K; \mathbb{F}_2)$ and compares it with the corresponding map for the mirror knot.

The search focused on prime knots with up to 17 crossings. While knots with up to 15 crossings and all 1,008,906 non-alternating 16-crossing prime knots were screened without success, the search proceeded to the 17-crossing census, which contains over 8 million prime knots.

Key Results
The paper presents Theorem 3.1, which identifies a specific prime knot, denoted as $K = 17nh0009090$ in the Regina census, that satisfies the conditions of Lipshitz and Sarkar's question:

• Homology Equivalence: For every pair of grading indices $(i, j)$, the integral Khovanov homology groups of $K$ and its mirror $mK$ are isomorphic as abelian groups ($Khi,j(K; \mathbb{Z}) \cong Khi,j(mK; \mathbb{Z})$), including all torsion components.
• Spectral Distinction: Over the field $\mathbb{F}_2$, the rank of the Steenrod square map $Sq^2: Kh_{-6,-9} \to Kh_{-4,-9}$ differs between the knot and its mirror. Specifically, the rank is 1 for $K$ and 0 for $mK$.
• Conclusion: Consequently, the quantum-graded Khovanov stable homotopy types of $K$ and $mK$ are not equivalent, despite their integral homology groups being identical.

The author also notes the existence of a composite (non-prime) example satisfying similar properties, detailed in Appendix B.

Significance and Claims
The paper claims to provide an affirmative answer to Lipshitz and Sarkar's Q2, demonstrating that the Khovanov spectrum is a strictly stronger invariant for detecting chirality than integral Khovanov homology. The author emphasizes that this is the first successful example found in a systematic computer-assisted search, marking the 17-crossing frontier as the first instance where such a discrepancy was observed.

The work highlights the utility of AI-driven tools in mathematical discovery, specifically in combinatorial searches and pattern recognition within large datasets like the Regina Census. The author frames the result as an existence theorem rather than a claim of absolute minimality, noting that the 17-crossing knot is the first verified example found but not necessarily the knot with the minimal crossing number for this property. The paper concludes that the obstruction to amphichirality provided by the Khovanov spectrum is indeed stronger than that provided by the homology alone.