Detecting essential surfaces as intersections in the character variety

TL;DR

This paper investigates the character variety of certain hyperbolic knots, revealing that intersection points between components encode topological features like Seifert surfaces, with implications for understanding knot invariants.

Contribution

It introduces a family of hyperbolic knots with character varieties having exactly two irreducible components and shows that their intersection points detect Seifert surfaces.

Findings

Intersection points are non-integral.

These points encode topological information.

The character variety structure aids in understanding knot surfaces.

Abstract

We describe a family of hyperbolic knots whose character variety contain exactly two distinct components of characters of irreducible representations. The intersection points between the components carry rich topological information. In particular, these points are non-integral and detect the Seifert surface.