Geometric structures and $PSL_2(\mathbb{C})$ representations of knot groups from knot diagrams

TL;DR

This paper introduces a diagram-based method to derive equations for the canonical component of the $PSL_2(C)$ representation variety of knot groups, simplifying the process without complex decompositions.

Contribution

It presents a novel, diagram-only algorithm for computing the representation variety equations, applicable to infinite families of knots, avoiding traditional polyhedral methods.

Findings

Provides explicit equations for hyperbolic structures of knots

Simplifies the computation process for representation varieties

Demonstrates the method on a family of braids depending on parameter n

Abstract

We describe a new method of producing equations for the canonical component of representation variety of a knot group into $PSL_2(\mathbb{C})$. Unlike known methods, this one does not involve any polyhedral decomposition or triangulation of the knot complement, and uses only a knot diagram satisfying a few mild restrictions. This gives a simple algorithm that can often be performed by hand, and in many cases, for an infinite family of knots at once. The algorithm yields an explicit description for the hyperbolic structures (complete or incomplete) that correspond to geometric representations of a hyperbolic knot. As an illustration, we give the formulas for the equations for the variety of closed alternating braids $(\sigma_1(\sigma_2)^{-1})^n$ that depend only on $n$.