The Alexander polynomial at prime powers

TL;DR

This paper explores the properties of the Alexander polynomial of a knot evaluated at prime power values, providing insights into its finite set structure and cardinality.

Contribution

It introduces a finite set associated with the Alexander polynomial at prime powers and analyzes its cardinality, offering new understanding of knot invariants at prime power evaluations.

Findings

Identifies a finite set related to the Alexander polynomial at prime powers.

Establishes the cardinality of this set as the polynomial's value at those points.

Provides theoretical insights into the behavior of knot invariants at prime power evaluations.

Abstract

When $K$ is a knot and $p \gg 0$ is a prime, we discuss a finite set whose cardinality is $\Delta_K(p^n)$, the value of the Alexander polynomial of $K$ at $p^n$.