Knot Floer homology and the unknotting number

Alishahi, Akram; Eftekhary, Eaman

doi:10.2140/gt.2020.24.2435

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Knot Floer homology and the unknotting number

Akram Alishahi and Eaman Eftekhary

Geometry & Topology 24 (2020) 2435–2469

DOI: 10.2140/gt.2020.24.2435

Abstract

Given a knot $K \subset S^{3}$ , 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 $𝔩 (K) \geq ν^{+} (K)$ , 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

Volume 24, issue 5 (2020)