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Representation varieties of twisted Hopf links

Ángel González-Prieto, Vicente Muñoz

Abstract

We study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the $SL_r(C)$-representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representation and character varieties.

Representation varieties of twisted Hopf links

Abstract

We study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the -representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representation and character varieties.
Paper Structure (18 sections, 7 theorems, 79 equations, 3 figures)

This paper contains 18 sections, 7 theorems, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Representation varieties and character varieties
  3. Hodge structures and $E$-polynomials
  4. Equivariant $E$-polynomial
  5. Twisted Hopf links
  6. The $\mathop{\mathrm{SL}}\nolimits_{r}(\mathbb{C})$-representation variety of the twisted Hopf link
  7. The combinatorial setting
  8. The geometric setting
  9. Representation variety of the twisted Hopf link of rank $2$
  10. Combinatorial setting
  11. Geometric setting
  12. Rank $2$ character variety of the twisted Hopf link
  13. Rank $3$ representation variety of the twisted Hopf link. Combinatorial setting
  14. Equivariant $E$-polynomials
  15. Rank $3$ representation variety of the twisted Hopf link. Geometric setting
  16. ...and 3 more sections

Key Result

Theorem 1

The $E$-polynomials of the $\mathop{\mathrm{SL}}\nolimits_r(\mathbb{C})$-representation variety of the twisted Hopf link $H_n$ with $n$ twists for ranks $r = 2,3$, are the following.

Figures (3)

  • Figure 1: The twisted Hopf link of $n$ twists.
  • Figure 2: The twisted Hopf link of $n$ twists with oriented strands.
  • Figure 3: A crossing of the $H_n$.

Theorems & Definitions (21)

  • Theorem
  • Theorem
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Example 4.1
  • Definition 4.2
  • ...and 11 more