Representation theory and the \(A\)-polynomial of a knot
Let \(X\) be a compact \(3\)-manifold with a single torus boundary component. In the paper [Invent. Math. 118, No. 1, 47-84 (1994; Zbl 0842.57013)], the authors (with \textit{M. Culler, H. Gillet}, and \textit{P. B. Shalen}) describe a certain affine algebraic curve which comes from the affine algebraic set of representations of the fundamental group \(\pi_1(X)\) into \(SL(2, C)\) in the language of algebraic geometry. A basis for the peripheral subgroup of \(\pi_1(X)\) gives a plane algebraic curve in \(C^2\) and it is determined by an integer polynomial in two variables. In case that \(X\) is the knot exterior of a knot \(K\) in \(S^3,\) the polynomial is an invariant of the knot \(K\), called the \(A\)-polynomial of \(K\). The paper under review surveys various aspects of the \(A\)-polynomials of knots and related results in the light of knot theoretical and geometrical significance. Especially, the authors discuss the properties of the \(A\)-polynomial of a knot which give information about the boundary slopes of incompressible surfaces in the exterior of a knot and Dehn-filled manifolds of knot complements, including numerous unresolved questions. This paper will undoubtedly be of value to researchers in the field as a unified introduction to many results.
scientific article; zbMATH DE number 1370413
scientific article