Hyperbolic knots with arbitrarily large torsion order in knot Floer homology

TL;DR

This paper demonstrates that hyperbolic knots, especially twisted torus knots, can have arbitrarily large torsion orders in knot Floer homology, extending known results from torus knots and using the Upsilon torsion function for analysis.

Contribution

It proves that hyperbolic knots can realize arbitrarily large torsion orders in knot Floer homology, a result previously known only for torus knots, and introduces a unified approach using the Upsilon torsion function.

Findings

Hyperbolic knots can have arbitrarily large torsion orders in knot Floer homology.

Most such knots are twisted torus knots.

A new infinite family of hyperbolic knots sharing a Upsilon torsion function is constructed.

Abstract

In knot Floer homology, there are two types of torsion order. One is the minimal power of the action of the variable $U$ to annihilate the $\mathbb{F}_2[U]$-torsion submodule of the minus version of knot Floer homology $\mathrm{HFK}^-(K)$. This is introduced by Juh\'{a}sz, Miller and Zemke, and denoted by $\mathrm{Ord}(K)$. The other, $\mathrm{Ord}'(K)$, introduced by Gong and Marengon, is similarly defined for the $\mathbb{F}_2[U]$-torsion submodule of the unoriented knot Floer homology $\mathrm{HFK}'(K)$. For both torsion orders, it is known that arbitrarily large values are realized by torus knots. In this paper, we prove that they can be realized by hyperbolic knots, most of which are twisted torus knots. Two torsion orders are argued in a unified way by using the Upsilon torsion function introduced by Allen and Livingston. We also give the first infinite family of hyperbolic knots which shares a common Upsilon torsion function.