Knots with large character varieties

TL;DR

This paper investigates knots with large-dimensional $ ext{SL}_2( ext{C})$-character varieties, introduces two diagrammatic methods to construct such knots, and conjectures that all Turk's head knots with odd parameters are $ ext{X}$-large.

Contribution

It introduces two new diagrammatic constructions for generating $ ext{X}$-large knots and conjectures all odd Turk's head knots are $ ext{X}$-large, expanding understanding of character varieties.

Findings

Most $ ext{X}$-large knots in tables explained topologically.

Constructed $ ext{X}$-large knots via split links and braids.

Conjecture all Turk's head knots with odd parameters are $ ext{X}$-large.

Abstract

We study knots whose $\mathrm{SL}_2(\mathbb{C})$-character varieties have a component of dimension greater than one. We call such knots $\mathcal{X}$-large and introduce two diagrammatic constructions that produce $\mathcal{X}$-large knots. The first construction uses split link diagrams and rational tangle replacements, providing a topological explanation for most $\mathcal{X}$-large knots observed in knot tables. The second construction is based on braids and orientation-reversing involutions, and is motivated by a detailed analysis of the knot $10_{123}$, also known as the Turk's head knot $Th(3,5)$. In particular, this approach applies to Turk's head knots $Th(p,q)$ with $p$ and $q$ odd, leading us to conjecture that all such knots are $\mathcal{X}$-large. In doing so, we also present a non-orientable analogue of Thurston's theorem giving a lower bound on the dimension of character varieties of non-orientable 3-manifolds.