On the First Singularity for the Upsilon Invariant of Algebraic Knots

TL;DR

This paper establishes a link between the first singularity of the Upsilon function for algebraic knots and their Puiseux characteristic sequence, providing improved bounds on cobordism genus over tau invariants.

Contribution

It introduces a method to determine the first singularity of the Upsilon function from the Puiseux sequence, enhancing understanding of algebraic knot invariants.

Findings

First singularity location determined by Puiseux sequence

Improved bounds on cobordism genus for algebraic knots

Enhanced understanding of Upsilon invariant behavior

Abstract

We show that the location of the first singularity of the Upsilon function of an algebraic knot is determined by the first term of its Puiseux characteristic sequence. In many cases this gives better bounds than the tau invariant on the genus of a cobordism between algebraic knots.