The $A$-Polynomial and Knot Volume

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.