Knot Floer homology and the unknotting number

TL;DR

This paper introduces new knot invariants derived from knot Floer homology that provide lower bounds on the unknotting number and related measures, with applications to knot theory problems.

Contribution

The authors construct the invariants l^-, l^+, and l(K) from knot Floer homology, establishing their effectiveness in bounding unknotting-related quantities and demonstrating their properties.

Findings

l(K) vanishes only for the unknot

l(K) is greater than or equal to ^-(K)

The difference l(K) - ^-(K) can be arbitrarily large

Abstract

Given a knot K in 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 is greater than or equal to the \nu^-(K). Moreover, the difference l(K)-\nu^-(K) can be arbitrarily large. We also present several applications towards bounding the unknotting number, the alteration number and the Gordian distance.