# A Fox-Milnor Theorem for Knots in a Thickened Surface

## TL;DR

This paper extends the classical Fox-Milnor Theorem to knots in thickened surfaces, using Milnor torsion, thereby generalizing key knot invariants to a broader topological setting.

## Contribution

It proves a Fox-Milnor Theorem for knots in thickened surfaces, expanding the classical result to a new context using Milnor torsion.

## Key findings

- Established a Fox-Milnor Theorem for knots in thickened surfaces.
- Connected Alexander polynomials of concordant knots in this setting.
- Utilized Milnor torsion to generalize classical knot invariants.

## Abstract

A knot in a thickened surface $K$ is a smooth embedding $K:S^1 \rightarrow \Sigma \times [0,1]$, where $\Sigma$ is a closed, connected, orientable surface. There is a bijective correspondence between knots in $S^2 \times [0,1]$ and knots in $S^3$, so one can view the study of knots in thickened surfaces as an extension of classical knot theory. An immediate question is if other classical definitions, concepts, and results extend or generalize to the study of knots in a thickened surface. One such famous result is the Fox Milnor Theorem, which relates the Alexander polynomials of concordant knots. We prove a Fox Milnor Theorem for concordant knots in a thickened surface by using Milnor torsion.