Non-triviality of the $M$-degree of the $A$-polynomial

TL;DR

This paper proves that the $A$-polynomial of any nontrivial knot in $S^3$ always has a nontrivial degree in the variable $M$, highlighting a fundamental property of knot invariants.

Contribution

It establishes a general proof that the $A$-polynomial's $M$-degree is nontrivial for all nontrivial knots in $S^3$, a previously unconfirmed property.

Findings

$A$-polynomial of nontrivial knots has nontrivial $M$-degree

Provides a universal proof for all nontrivial knots in $S^3$

Strengthens understanding of knot invariants

Abstract

This note gives a proof that the $A$-polynomial of any nontrivial knot in $S^3$ has nontrivial $M$-degree.