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Quantum traces for $\mathrm{SL}_n(\mathbb{C})$: The case $n=3$

Daniel C. Douglas

Abstract

We generalize Bonahon-Wong's $\mathrm{SL}_2(\mathbb{C})$-quantum trace map to the setting of $\mathrm{SL}_3(\mathbb{C})$. More precisely, given a non-zero complex parameter $q=e^{2 πi \hbar}$, we associate to each isotopy class of framed oriented links $K$ in a thickened punctured surface $\mathfrak{S} \times (0, 1)$ a Laurent polynomial $\mathrm{Tr}_λ^q(K) = \mathrm{Tr}_λ^q(K)(X_i^q)$ in $q$-deformations $X_i^q$ of the Fock-Goncharov $\mathcal{X}$-coordinates for higher Teichmüller space. This construction depends on a choice $λ$ of ideal triangulation of the surface $\mathfrak{S}$. Along the way, we propose a definition for a $\mathrm{SL}_n(\mathbb{C})$-version of this invariant.

Quantum traces for $\mathrm{SL}_n(\mathbb{C})$: The case $n=3$

Abstract

We generalize Bonahon-Wong's -quantum trace map to the setting of . More precisely, given a non-zero complex parameter , we associate to each isotopy class of framed oriented links in a thickened punctured surface a Laurent polynomial in -deformations of the Fock-Goncharov -coordinates for higher Teichmüller space. This construction depends on a choice of ideal triangulation of the surface . Along the way, we propose a definition for a -version of this invariant.

Paper Structure

This paper contains 72 sections, 7 theorems, 136 equations, 41 figures.

Table of Contents

  1. Introduction
  2. Classical trace polynomials for SLn
  3. Topological setup
  4. Discrete triangle
  5. Dotted ideal triangulations
  6. Classical polynomial algebra
  7. Elementary edge and triangle matrices
  8. Local monodromy matrices
  9. Definition of the SLn-classical trace polynomials
  10. Relation to Fock--Goncharov theory
  11. Quantum matrices for SLn
  12. Quantum tori, matrix algebras, and the Weyl quantum ordering
  13. Quantum tori
  14. Matrix algebras
  15. Weyl quantum ordering
  16. ...and 57 more sections

Key Result

Theorem \oldthetheorem

Fix a $(2*3^2)$-root $\omega^{1/2} = q^{1/(2*3^2)}=q^{1/18} \in \mathbb{C} - \{ 0 \}$. For each ideal triangulation $\lambda$ of the punctured surface $\mathfrak{S}$ (with possibly non-empty boundary), there exists a function such that if $\omega^{1/2}=1$, then for every blackboard-framed oriented link $K$ whose components $K_1, K_2, \dots, K_\ell$ project to immersed closed oriented curves $\gam

Figures (41)

  • Figure 1: HOMFLYPT skein relation.
  • Figure 2: Unknot skein relation.
  • Figure 3: Framing skein relations.
  • Figure 4: Ideal triangulations ($\partial \mathfrak{S} = \varnothing$).
  • Figure 5: Discrete triangle ($n=5$).
  • ...and 36 more figures

Theorems & Definitions (45)

  • Conjecture \oldthetheorem: $\mathrm{SL}_n(\mathbb{C})$-quantum trace map
  • Theorem \oldthetheorem: Theorem \ref{['thm:second-theorem']}, $\mathrm{SL}_3(\mathbb{C})$-quantum trace polynomials
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem: FockIHES06, $\mathrm{SL}_n$-classical trace polynomials
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • ...and 35 more