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We prove that Khovanov homology with coefficients in\Z/2\Zdetects the(2,5)torus knot. Our proof makes use of a wide range of deep tools in Floer homology, Khovanov homology, and Khovanov homotopy. We combine these tools with classical results on the dynamics of surface homeomorphisms to reduce the detection question to a problem about mutually braided unknots, which we then solve with computer assistance.
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Rank bounds in link Floer homology and detection results
Viewing the BRAID invariant as a generator of link Floer homology, we generalize work of Baldwin–Vela-Vick to obtain rank bounds on the next-to-top grading of knot Floer homology. These allow us to classify links with knot Floer homology of rank at most eight and prove a variant of a classification of links with Khovanov homology of low rank due to Xie–Zhang. In another direction, we use a variant of Ozsváth–Szabó's classification ofE_2collapsed\mathbb{Z}\oplus\mathbb{Z}filtered chain complexes to show that knot Floer homology detectsT(2,8)andT(2,10). Combining these techniques with the spectral sequences of Batson–Seed, Dowlin, and Lee, we can show that Khovanov homology likewise detectsT(2,8)andT(2,10).
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- PAR ID:
- 10597664
- Publisher / Repository:
- Dey-QT
- Date Published:
- Journal Name:
- Quantum Topology
- ISSN:
- 1663-487X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4171/QT/197
