The ${\rm SL}(2,\mathbb{C})$-character variety of a Montesinos knot

Abstract

For each Montesinos knot K, we find a simple method to determine the SL(2, C)-character variety, and show that it can be decomposed as X0(K) X1(K) X2(K) X (K), where X0(K) consists of trace-free characters, X1(K) consists of characters of "connected sums" of representations of the constituent rational links, X2(K) is a high-genus algebraic curve, and X (K) generically consists of finitely many points.

The adjoint Reidemeister torsion for the connected sum of knots

“…The simplest one might be (the knot exterior of) the connected sum of knots. We refer to [CL96,PP13,Che21] for other examples. Two immediate problems when we consider Conjecture 1.1 for such 3-manifolds are that (P1) the adjoint Reidemeister torsion is not defined for a component of dimension ≥ 2; (P2) the sum in the equation (1) does not make sense as the level set X c µ (M ) is no longer finite.…”