On an integral version of the Rasmussen invariant

TL;DR

This paper introduces an integral version of the Rasmussen s-invariant, explores its relation to existing invariants, and demonstrates its effectiveness in providing sharper bounds on knot slice genus compared to field-based invariants.

Contribution

It defines a new integral s-invariant, establishes its connection to field-based invariants, and shows it yields improved lower bounds on the slice genus for certain knots.

Findings

The integral s-invariant provides better lower bounds than field-based invariants for some knots.

The new invariant relates to the Lipshitz-Sarkar refinement involving Steenrod squares.

Examples demonstrate the invariant's effectiveness in knot genus estimation.

Abstract

We define a Rasmussen $s$-invariant over the coefficient ring of the integers, and show how it is related to the $s$-invariants defined over a field. A lower bound for the slice genus of a knot arising from it is obtained, and we give examples of knots for which this lower bound is better than all lower bounds coming from the $s$-invariants over fields. We also compare it to the Lipshitz-Sarkar refinement related to the first Steenrod square.