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Involutive Khovanov homology and equivariant knots (2404.08568v4)
Published 12 Apr 2024 in math.GT
Abstract: For strongly invertible knots, we define an involutive version of Khovanov homology, and from it derive a pair of integer-valued invariants $(\underline{s}, \bar{s})$, which is an equivariant version of Rasmussen's $s$-invariant. Using these invariants, we reprove that the infinite family of knots $J_n$ introduced by Hayden each admits exotic pairs of slice disks. Our construction is intended to give a Khovanov-theoretic analogue of the formalism given by Dai, Mallick and Stoffregen in involutive knot Floer theory.
- Akram Alishahi “Unknotting number and Khovanov homology” In Pacific J. Math. 301.1, 2019, pp. 15–29 URL: https://doi.org/10.2140/pjm.2019.301.15
- Dror Bar-Natan “Khovanov’s homology for tangles and cobordisms” In Geom. Topol. 9, 2005, pp. 1443–1499 DOI: 10.2140/gt.2005.9.1443
- “Equivariant 4-genera of strongly invertible and periodic knots” In J. Topol. 15.3, 2022, pp. 1635–1674
- Tim D. Cochran and Eamonn Tweedy “Positive links” In Algebr. Geom. Topol. 14.4, 2014, pp. 2259–2298 DOI: 10.2140/agt.2014.14.2259
- Irving Dai, Abhishek Mallick and Matthew Stoffregen “Equivariant knots and knot Floer homology” In J. Topol. 16.3, 2023, pp. 1167–1236 DOI: 10.1112/topo.12312
- C.McA. Gordon “On the higher-dimensional Smith conjecture” In Proc. London Math. Soc. (3) 29, 1974, pp. 98–110 DOI: 10.1112/plms/s3-29.1.98
- Kyle Hayden “Corks, covers, and complex curves”, 2021 arXiv:2107.06856 [math.GT]
- “Khovanov homology and exotic surfaces in the 4-ball” In J. Reine Angew. Math. 809, 2024, pp. 217–246 DOI: 10.1515/crelle-2024-0001
- Kristen Hendricks, Ciprian Manolescu and Ian Zemke “A connected sum formula for involutive Heegaard Floer homology” In Selecta Math. (N.S.) 24.2, 2018, pp. 1183–1245 DOI: 10.1007/s00029-017-0332-8
- “A cobordism realizing crossing change on 𝔰𝔩2𝔰subscript𝔩2\mathfrak{sl}_{2}fraktur_s fraktur_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT tangle homology and a categorified Vassiliev skein relation” In Topology Appl. 296, 2021, pp. Paper No. 107646\bibrangessep31 DOI: 10.1016/j.topol.2021.107646
- Mikhail Khovanov “A categorification of the Jones polynomial” In Duke Math. J. 101.3, 2000, pp. 359–426 DOI: 10.1215/S0012-7094-00-10131-7
- Artem Kotelskiy, Liam Watson and Claudius Zibrowius “Immersed curves in Khovanov homology” arXiv, 2019 DOI: 10.48550/ARXIV.1910.14584
- Eun Soo Lee “An endomorphism of the Khovanov invariant” In Adv. Math. 197.2, 2005, pp. 554–586 DOI: 10.1016/j.aim.2004.10.015
- Lukas Lewark “The Rasmussen invariant of arborescent and of mutant links”, 2009
- “Khovanov homology of strongly invertible knots and their quotients”, 2022 arXiv:2203.13895 [math.GT]
- “A refinement of Khovanov homology” In Geom. Topol. 25.4, 2021, pp. 1861–1917 DOI: 10.2140/gt.2021.25.1861
- “A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds” In Duke Math. J. 172.2, 2023, pp. 231–311 DOI: 10.1215/00127094-2022-0039
- “The Smith conjecture” Papers presented at the symposium held at Columbia University, New York, 1979 112, Pure and Applied Mathematics Academic Press, Inc., Orlando, FL, 1984, pp. xv+243
- Lisa Piccirillo “The Conway knot is not slice” In Ann. of Math. (2) 191.2, 2020, pp. 581–591 DOI: 10.4007/annals.2020.191.2.5
- Jacob Rasmussen “Khovanov homology and the slice genus” In Invent. Math. 182.2, 2010, pp. 419–447 DOI: 10.1007/s00222-010-0275-6
- Makoto Sakuma “On strongly invertible knots” In Algebraic and topological theories (Kinosaki, 1984) Kinokuniya, Tokyo, 1986, pp. 176–196
- Taketo Sano “A description of Rasmussen’s invariant from the divisibility of Lee’s canonical class” In J. Knot Theory Ramifications 29.6, 2020, pp. 2050037\bibrangessep39 DOI: 10.1142/S0218216520500376
- “A family of slice-torus invariants from the divisibility of Lee classes”, 2023 arXiv:2211.02494 [math.GT]
- Alexander N. Shumakovitch “Torsion of Khovanov homology” In Fund. Math. 225.1, 2014, pp. 343–364 DOI: 10.4064/fm225-1-16
- Michael Snape “Homological invariants of strongly invertible knots” University of Glasgow, 2018
- Paul Turner “Khovanov homology and diagonalizable Frobenius algebras” In J. Knot Theory Ramif. 29.01 World Scientific Publishing Co., 2020, pp. 1950095
- Friedhelm Waldhausen “Über Involutionen der 3333-Sphäre” In Topology 8, 1969, pp. 81–91 DOI: 10.1016/0040-9383(69)90033-0
- Liam Watson “Khovanov homology and the symmetry group of a knot” In Adv. Math. 313, 2017, pp. 915–946 DOI: 10.1016/j.aim.2017.04.003
- S. Wehrli “A spanning tree model for Khovanov homology” In J. Knot Theory Ramifications 17.12, 2008, pp. 1561–1574 DOI: 10.1142/S0218216508006762
- Yuval Wigderson “The Bar-Natan theory splits” In J. Knot Theory Ramifications 25.4, 2016, pp. 1650014\bibrangessep19 DOI: 10.1142/S0218216516500140
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