Gonality and genus of canonical components of character varieties

TL;DR

This paper investigates the algebraic complexity of canonical components of character varieties of 3-manifolds after Dehn filling, showing bounded gonality and genus, with explicit calculations for certain knots.

Contribution

It establishes bounds on gonality and genus of character varieties post-Dehn filling and computes gonality for specific knot examples, revealing unbounded gonality in some cases.

Findings

Gonality of character varieties is bounded independently of filling parameters.

Bounds on genus depend on the filling parameter r.

Double twist knots exhibit arbitrarily large gonality.

Abstract

Let M be a two cusped hyperbolic 3-manifold and let M(r) be the result of r Dehn filling of a fixed cusp of M. We study canonical components of the SL(2,C) character varieties of M(r). We show that the gonality of these sets is bounded, independent of the filling parameter. We also obtain bounds, depending on r, for the genus of these sets. We compute the gonality for the double twist knots, demonstrating canonical components with arbitrarily large gonality.