Closed surfaces and character varieties

TL;DR

This paper enhances the use of character variety techniques to detect essential surfaces and boundary slopes in knot manifolds, introducing new algebraic and number theoretic invariants.

Contribution

It introduces module structures on coordinate rings to identify boundary slopes and closed surfaces, advancing the algebraic tools in character variety analysis.

Findings

Module structures help detect boundary slopes.

New invariants for character varieties are developed.

Closed surfaces detection is improved.

Abstract

The powerful character variety techniques of Culler and Shalen can be used to find essential surfaces in knot manifolds. We show that module structures on the coordinate ring of the character variety can be used to identify detected boundary slopes as well as when closed surfaces are detected. This approach also yields new number theoretic invariants for the character varieties of knot manifolds.