An Upsilon torsion function for knot Floer homology

TL;DR

This paper introduces the Upsilon torsion function, a new invariant in knot Floer homology that interpolates existing torsion invariants and provides bounds on knot cobordisms and Gordian distance.

Contribution

It defines a one-parameter family of torsion invariants called the Upsilon torsion function, connecting previous invariants and offering new obstructions in knot theory.

Findings

Defines the Upsilon torsion function as a piecewise linear function on [0,2]

Interpolates between Juhasz-Miller-Zemke and Gong-Marengon invariants

Provides new bounds on knot cobordisms and Gordian distance

Abstract

Heegaard Floer theory produces chain complexes associated to knots. Viewed as modules over polynomial rings, such complexes yield torsion invariants that offer constraints on cobordisms between knots. For instance, Juhasz, Miller and Zemke used torsion invariants to bound the number of local maxima and minima in cobordisms between pairs of knots. Gong and Marengon defined a related torsion invariant and used it to study nonorientable knot cobordisms. In this paper we define a one parameter family of Heegaard Floer torsion invariants that yields a piecewise linear function defined on the interval [0,2]. We call this the Upsilon torsion function; it is closely related to the Heegaard Floer Upsilon function defined by Ozsvath, Stipsicz and Szabo. In a natural way, this Upsilon torsion function interpolates between the Juhasz-Miller-Zemke invariant and the Gong-Marengon invariant. In addition to bounding the number of local maxima and minima in knot cobordisms, the Upsilon torsion function provides new obstructions related to the Gordian distance between knots.