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

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.