Volume 24, issue 5 (2020)

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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.

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