High-dimensional components of ${\rm SL}(2,\mathbb{C})$-character varieties of prime knots

TL;DR

This paper investigates the structure of ${ m SL}(2,bC)$-character varieties of prime knots, providing conditions for higher-dimensional components, improving previous bounds, and answering a question about their dimensions.

Contribution

It offers new criteria for the existence of higher-dimensional components in character varieties and improves lower bounds for their dimensions, advancing understanding of knot invariants.

Findings

Established sufficient conditions for higher-dimensional components.

Provided a lower bound for the dimension of these components.

Extended results to Montesinos knots and addressed a 2018 question.

Abstract

For a prime knot $K$, we give sufficient conditions for the existence of a component $\mathcal{C}$ of the irreducible ${\rm SL}(2,\mathbb{C})$-character variety of $K$ with $\dim\mathcal{C}>1$, and give a lower bound for $\dim\mathcal{C}$. Specifically, we improve a result of Paoluzzi and Porti on Montesinos knots, and positively answer a question posed by Culler and Dunfield in 2018.