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Knot Floer homology and the unknotting number

Akram Alishahi and Eaman Eftekhary

Geometry & Topology 24 (2020) 2435–2469
Abstract

Given a knot K S3, let u(K) (respectively, u+(K)) denote the minimum number of negative (respectively, positive) crossing changes among all unknotting sequences for K. We use knot Floer homology to construct the invariants 𝔩(K), 𝔩+(K) and 𝔩(K), which give lower bounds on u(K), u+(K) and the unknotting number u(K), respectively. The invariant 𝔩(K) only vanishes for the unknot, and satisfies 𝔩(K) ν+(K), while the difference 𝔩(K) ν+(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.

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Keywords
knot, unknotting number, knot Floer homology, torsion
Mathematical Subject Classification 2010
Primary: 57M27
References
Publication
Received: 10 November 2018
Revised: 19 December 2019
Accepted: 10 March 2020
Published: 29 December 2020
Proposed: Ciprian Manolescu
Seconded: András I Stipsicz, Ian Agol
Authors
Akram Alishahi
Department of Mathematics
University of Georgia
Athens, GA
United States
Eaman Eftekhary
School of Mathematics
Institute for Research in Fundamental Sciences (IPM)
Tehran
Iran