Non-standard components of the character variety for a family of Montesinos knots

TL;DR

This paper demonstrates that certain Montesinos knots have character varieties with arbitrarily many irreducible components of various dimensions, including components with non-constant meridian trace, revealing complex algebraic structures.

Contribution

It introduces new examples of Montesinos knots with richly structured character varieties, expanding understanding of their algebraic and geometric properties.

Findings

Existence of knots with many irreducible components in their character varieties

Components can have any dimension between 1 and n-2

Some components have non-constant meridian trace

Abstract

We show that there are Montesinos knots with $n+1\geq 4$ tangles whose character varieties contain arbitrarily many irreducible components of dimension $d$ for any $1\le d\le n-2$. Moreover, these irreducible components can be chosen so that the trace of the meridian is non-constant.