The $A$-Polynomial and Knot Volume

TL;DR

This paper explores a potential link between the $A$-polynomial of knots and their hyperbolic volume, proposing a conjecture that connects algebraic properties of the polynomial with geometric volume, supported by initial evidence.

Contribution

It conjectures a precise relationship between the $A$-polynomial factors and hyperbolic volume, and provides initial results supporting the forward direction of this conjecture.

Findings

Knots with zero hyperbolic volume have $A$-polynomials with factors as sums of two monomials.

The paper shows the forward implication of the conjecture.

Evidence suggests the converse may also hold for certain knots.

Abstract

In this paper, we conjecture a connection between the $A$-polynomial of a knot in $\mathbb{S}^{3}$ and the hyperbolic volume of its exterior $\mathcal{M}_{K}$ : the knots with zero hyperbolic volume are exactly the knots with an $A$-polynomial where every irreducible factor is the sum of two monomials in $L$ and $M$. Herein, we show the forward implication and examine cases that suggest the converse may also be true. Since the $A$-polynomial of hyperbolic knots are known to have at least one irreducible factor which is not the sum of two monomials in $L$ and $M$, this paper considers satellite knots which are graph knots and some with positive hyperbolic volume.