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Award ID contains: 1952707

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  1. ; (, Forum of Mathematics, Pi)
    Abstract We prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots and that HOMFLY homology detects infinitely many knots; these theories were previously known to detect a mere six knots, all fibered. These results rely on our main technical theorem, which gives a complete classification of genus-1 knots in the 3-sphere whose knot Floer homology in the top Alexander grading is 2-dimensional. We discuss applications of this classification to problems in Dehn surgery which are carried out in two sequels. These include a proof that$$0$$-surgery characterizes infinitely many knots, generalizing results of Gabai from his 1987 resolution of the Property R Conjecture. 
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    Free, publicly-accessible full text available January 1, 2026
  2. ; ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract We prove that the knot Floer complex of a fibered knot detects whether the monodromy of its fibration is right-veering.In particular, this leads to a purely knot Floer-theoretic characterization of tight contact structures, by the work of Honda–Kazez–Matić.Our proof makes use of the relationship between the Heegaard Floer homology of mapping tori and the symplectic Floer homology of area-preserving surface diffeomorphisms.We describe applications of this work to Dehn surgeries and taut foliations. 
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  3. ; (, Journal of the London Mathematical Society)
    Abstract We prove that all rational slopes are characterizing for the knot , except possibly for positive integers. Along the way, we classify the Dehn surgeries on knots in that produce the Brieskorn sphere , and we study knots on which large integral surgeries are almost L‐spaces. 
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  4. ; ; (, Journal of the European Mathematical Society)
    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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    Free, publicly-accessible full text available April 9, 2026
  5. ; (, Proceedings of Symposia in Pure Mathematics)
    Accepted Manuscript Full Text Available
  6. ; ; ; (, Geometry & Topology)
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  7. ; (, Mathematical Research Letters)
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  8. ; ; (, Compositio Mathematica)
    Suppose$$\mathcal {H}$$is an admissible Heegaard diagram for a balanced sutured manifold$$(M,\gamma )$$. We prove that the number of generators of the associated sutured Heegaard Floer complex is an upper bound on the dimension of the sutured instanton homology$$\mathit {SHI}(M,\gamma )$$. It follows, in particular, that strong L-spaces are instanton L-spaces. 
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  9. ; (, Annales Henri Lebesgue)
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