REMARKS ON THE A-POLYNOMIAL OF A KNOT

Abstract

This paper reviews the two variable polynomial invariant of knots defined using representations of the fundamental group of the knot complement into [Formula: see text]. The slopes of the sides of the Newton polygon of this polynomial are boundary slopes of incompressible surfaces in the knot complement. The polynomial also contains information about which surgeries are cyclic, and about the shape of the cusp when the knot is hyperbolic. We prove that at least some mutants have the same polynomial, and that most untwisted doubles have non-trivial polynomial. We include several open questions.