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On the local structure of the SL(n,C)-representation variety of knot groups

Michael Heusener, Leila Ben Abdelghani

TL;DR

The authors analyze the local algebraic structure of the SL(n,$\mathbb C$) representation variety of knot groups near diagonal abelian representations. They construct a solvable upper-triangular deformation $\rho^{(n)}_D$ whose orbit closure contains the diagonal representation $\rho_D$, and prove that $\rho_D$ lies on a unique irreducible component $R^{(n)}_D$ of dimension $n^2+n-2$ that also contains irreducibles; they further compute the Zariski tangent and quadratic cones at $\rho_D$ and show these cones coincide, yielding a reduced tangent cone and smooth points on the relevant components. Using cohomological tools, they determine the structure and dimensions of nearby components, including intersections with the abelian component $R_n(\mathbb{Z})$, and exhibit how $\rho_D$ can be deformed into irreducibles. The local picture of the character variety is then illuminated via Luna's slice theorem, giving an etale-local model around the abelian character $\chi_D$ and clarifying smoothness and transverse intersections between components. The paper also provides concrete examples (torus knots, quadratic Alexander polynomial cases, trefoil, etc.) to illustrate the component structure and the deformations available in this setting.

Abstract

We study the local structure of the representation variety of a knot group into SL(n,C) at certain diagonal representations. In particular we determine the tangent cone of the representation variety at these diagonal representations, and show that the latter can be deformed into irreducible representations. Furthermore, we use Luna's slice theorem to analyze the local structure of the character variety.

On the local structure of the SL(n,C)-representation variety of knot groups

TL;DR

The authors analyze the local algebraic structure of the SL(n,) representation variety of knot groups near diagonal abelian representations. They construct a solvable upper-triangular deformation whose orbit closure contains the diagonal representation , and prove that lies on a unique irreducible component of dimension that also contains irreducibles; they further compute the Zariski tangent and quadratic cones at and show these cones coincide, yielding a reduced tangent cone and smooth points on the relevant components. Using cohomological tools, they determine the structure and dimensions of nearby components, including intersections with the abelian component , and exhibit how can be deformed into irreducibles. The local picture of the character variety is then illuminated via Luna's slice theorem, giving an etale-local model around the abelian character and clarifying smoothness and transverse intersections between components. The paper also provides concrete examples (torus knots, quadratic Alexander polynomial cases, trefoil, etc.) to illustrate the component structure and the deformations available in this setting.

Abstract

We study the local structure of the representation variety of a knot group into SL(n,C) at certain diagonal representations. In particular we determine the tangent cone of the representation variety at these diagonal representations, and show that the latter can be deformed into irreducible representations. Furthermore, we use Luna's slice theorem to analyze the local structure of the character variety.
Paper Structure (28 sections, 32 theorems, 166 equations)

This paper contains 28 sections, 32 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Plan of the article.
  3. Notations and background results
  4. Group cohomology
  5. Affine algebraic schemes
  6. Curve tangents and quadratic cone
  7. Representation scheme and variety
  8. The Alexander module
  9. Proof of the first part of Theorem \ref{['thm:main1']}
  10. The triangular representation
  11. Assumption.
  12. Cohomolgical calculations
  13. Irreducible $\mathrm{SL}(n,\mathbb{C})$ representations
  14. The diagonal representation
  15. The Zariski tangent space at the diagonal representation
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $K$ be a knot in a three-dimensional integer homology sphere, and let $D=D(\lambda_1,\ldots,\lambda_n)\in\mathrm{SL}(n,\mathbb{C})$ be a regular diagonal matrix. If all quotients $\alpha_i=\lambda_i/\lambda_{i+1}$, $i=1,\ldots,n-1$, are simple roots of the Alexander polynomial $\Delta_K$ and if

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 56 more