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

Kathleen L. Petersen, Anastasiia Tsvietkova

TL;DR

The paper tackles the problem of determining the canonical component of the $PSL_2(\mathbb{C})$-representation variety of knot groups directly from knot diagrams. It introduces a diagram-driven framework using taut diagrams and banana geometry to encode meridians, longitudes, and crossing/peripheral arcs as explicit $PSL_2(\mathbb{C})$ matrices, and it proves a normalization that reduces these to simple edge, region, and commuting relations. The main contribution is an explicit algorithm that computes the geometric component (and lifts to $SL_2(\mathbb{C})$) without triangulations, including practical shortcuts for bigon and 3-sided regions, with demonstrated applications to the figure-eight knot and an infinite family of closed 3-braids, yielding cusp shapes and the $A$-polynomial. This diagram-based method provides a direct path to understanding hyperbolic structures and their deformations, potentially enabling hand calculations and broader explorations of knot representation varieties. The work thus offers a significant, triangulation-free tool for studying geometric representations and their associated invariants across wide knot families.

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 $(σ_1(σ_2)^{-1})^n$ that depend only on $n$.

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

TL;DR

The paper tackles the problem of determining the canonical component of the -representation variety of knot groups directly from knot diagrams. It introduces a diagram-driven framework using taut diagrams and banana geometry to encode meridians, longitudes, and crossing/peripheral arcs as explicit matrices, and it proves a normalization that reduces these to simple edge, region, and commuting relations. The main contribution is an explicit algorithm that computes the geometric component (and lifts to ) without triangulations, including practical shortcuts for bigon and 3-sided regions, with demonstrated applications to the figure-eight knot and an infinite family of closed 3-braids, yielding cusp shapes and the -polynomial. This diagram-based method provides a direct path to understanding hyperbolic structures and their deformations, potentially enabling hand calculations and broader explorations of knot representation varieties. The work thus offers a significant, triangulation-free tool for studying geometric representations and their associated invariants across wide knot families.

Abstract

We describe a new method of producing equations for the canonical component of representation variety of a knot group into . 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 that depend only on .
Paper Structure (32 sections, 17 theorems, 72 equations, 12 figures)

This paper contains 32 sections, 17 theorems, 72 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Outline
  4. Background: Geometric Representations and Characters
  5. Lifting representations from $\text{PSL}_2(\mathbb{C})$ to $\text{SL}_2(\mathbb{C})$
  6. Geometric Representations and Invariant surfaces in $\mathbb{H}^3$
  7. Zariski Closure.
  8. Geometric Set-up: Peripheral and Crossing Arcs
  9. Suitable knot diagrams.
  10. Representation of a meridian.
  11. Representation of peripheral arcs
  12. Representation of crossing arcs
  13. Geometric Set-Up: Paths
  14. Normalizing Wirtinger Generators
  15. Degenerate Geometries.
  16. ...and 17 more sections

Key Result

Lemma 3.6

Up to conjugation, we can take $\rho(\mu)={\mathcal{M}}$. Further, we can specify that $|m|\geq 1$, and if $|m|=1$ then $\arg(m)\leq \pi$. This uniquely determines ${\mathcal{M}}$.

Figures (12)

  • Figure 1: Two bananas that are not horospheres.
  • Figure 2: Right: A path on a thickened knot, consisting of a crossing arc (in red) and a peripheral arc (in green). Left: A region of a knot diagram with five edges (in black). Five crossing arcs (in red) are also depicted.
  • Figure 3: The arcs $\mu_i, \mu_{i+1}$ on bananas $B_i, B_{i+1}$ are preimages of meridians.
  • Figure 4: The arc $\tilde{\gamma}$ from $\tilde{z}_1$ to $\tilde{z}_2$ weaves through banana $B$.
  • Figure 5: A lift of $\beta$ to $H$, with $z_1\neq z_2$ on the left, and $z_1=z_2$ on the right.
  • ...and 7 more figures

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • Remark 3.5
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • proof
  • ...and 44 more