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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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This content will become publicly available on April 9, 2026
Khovanov homology and the cinquefoil
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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- Award ID(s):
- 1952707
- PAR ID:
- 10598955
- Publisher / Repository:
- European Mathematical Society
- Date Published:
- Journal Name:
- Journal of the European Mathematical Society
- Volume:
- 27
- Issue:
- 6
- ISSN:
- 1435-9855
- Page Range / eLocation ID:
- 2443 to 2465
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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