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A Non-Abelian Generalization of the Alexander Polynomial from Quantum $\mathfrak{sl}_3$

Matthew Harper

Abstract

One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. We generalize this construction to define a link invariant $Δ_{\mathfrak{g}}$ for any semisimple Lie algebra $\mathfrak{g}$ of rank $n$, taking values in $n$-variable Laurent polynomials. Focusing on the case $\mathfrak{g}=\mathfrak{sl}_3$, we establish a direct relation between $Δ_{\mathfrak{sl}_3}$ and the Alexander polynomial. We show that certain parameter evaluations of $Δ_{\mathfrak{sl}_3}$ recover the Alexander polynomial on knots, despite the $R$-matrix not satisfying the Alexander-Conway skein relation at these points. We tabulate $Δ_{\mathfrak{sl}_3}$ for all knots up to seven crossings and various other examples, including the Kinoshita-Terasaka knot and Conway knot mutant pair which are distinguished by this invariant.

A Non-Abelian Generalization of the Alexander Polynomial from Quantum $\mathfrak{sl}_3$

Abstract

One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum at a fourth root of unity. We generalize this construction to define a link invariant for any semisimple Lie algebra of rank , taking values in -variable Laurent polynomials. Focusing on the case , we establish a direct relation between and the Alexander polynomial. We show that certain parameter evaluations of recover the Alexander polynomial on knots, despite the -matrix not satisfying the Alexander-Conway skein relation at these points. We tabulate for all knots up to seven crossings and various other examples, including the Kinoshita-Terasaka knot and Conway knot mutant pair which are distinguished by this invariant.

Paper Structure

This paper contains 29 sections, 35 theorems, 72 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. An overview of quantum group invariants
  3. Main results
  4. Tabulation of the invariant
  5. Relation to other invariants
  6. Further questions
  7. Structure of paper
  8. Restricted quantum sl_3
  9. Representations of U_z(sl_3)
  10. Induced representations
  11. Representations W_i(t)
  12. Representations W_{12}(t)
  13. Tensor product decompositions
  14. Unrolled restricted quantum sl_3 and braiding
  15. The unrolled quantum group
  16. ...and 14 more sections

Key Result

Theorem 1.1

The invariant $\Delta_{\mathfrak{sl}_3}$ has the following properties:

Figures (11)

  • Figure 1: Sketch of the curves $\mathscr{X}_\alpha\subset (\mathbb{C}^\times)^2$: $\mathscr{X}_1=\bigl\{(t_1,t_2)\mid t_1^2=1\bigr\}$, $\mathscr{X}_2=\bigl\{(t_1,t_2)\mid t_2^2=1\bigr\}$, $\mathscr{X}_{12}=\bigl\{(t_1,t_2)\mid (t_1t_2)^2=-1\bigr\}$. Each point on a unique $\mathscr{X}_\alpha$ determines an evaluation to the Alexander polynomial and is a highest weight of $V({{\text{\boldmath{$t$}}}})$ with irreducible subrepresentation $W_\alpha({{\text{\boldmath{$t$}}}})$.
  • Figure 2: The values of $\Delta_{\mathfrak{sl}_3}$ on the mutant pair $\mathsf{11_{n34}}$ and $\mathsf{11_{n42}}$.
  • Figure 3: The value of $\Delta_{\mathfrak{sl}_3}$ on the untwisted Whitehead double of $\mathsf{3_{1}}$.
  • Figure 4: The action of $\overline{U}$ on the weight spaces of $V({{\text{\boldmath{$t$}}}})$.
  • Figure 5: Reducible $V({{\text{\boldmath{$t$}}}})$ with subrepresentation $W_i({{\text{\boldmath{$t$}}}} \cdot {{\text{\boldmath{$\sigma_i$}}}})$.
  • ...and 6 more figures

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['thm:plugin']}
  • Theorem 1.3: Theorem \ref{['thm:smallAlexander']}
  • Theorem 1.4: Theorem \ref{['thm:torusknots']}
  • Conjecture 1.6
  • Conjecture 1.7
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • ...and 53 more