Research ArticleNo Access

The $A$ -polynomial and knot volume

Department of Mathematics and Physics, Ohio Northern University, Ada, OH 45810, USA

https://doi.org/10.1142/S0218216525500026Cited by:0 (Source: Crossref)

Abstract

In this paper, we conjecture a connection between the $A$ -polynomial of a knot in $𝕊^{3}$ and the hyperbolic volume of its exterior $ℳ_{K}$ : 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.

Keywords:

AMSC: 57M25, 57M27

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