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Affine representations of rational and pretzel knots

Javier Martínez

TL;DR

The paper develops a TQFT-based framework that realizes rational and pretzel knot groups as ${\textup{AGL}_{1}}(\mathbb{C})$-representations through spans of vector spaces, enabling explicit computations of Alexander invariants for these families. By constructing and analyzing the functors ${\mathcal{A}}$ and ${\mathcal{A}}_{t}$ on tangles, the authors derive closed-form Alexander polynomials expressed via continuants and quantum integers, and establish zeros-location results (e.g., unit-circle zeros for many pretzel cases). A detailed treatment of 2-bridge knots shows how Alexander polynomials arise as equalizers of tangle-maps, while pretzel knots are handled through a block-structured, cyclic tridiagonal framework yielding determinant formulas. The specialization ${\mathcal{A}}_{-1}$ connects to rational-tangle colorings, producing a geometric description of rational tangles as a rational curve in $\mathrm{Gr}(2,4)$ and linking coloring fractions to Grassmannian slopes. Overall, the approach unifies representation-theoretic, combinatorial, and geometric perspectives on Alexander-type invariants for a broad class of knots and tangles with concrete, calculable expressions.

Abstract

We construct and study representations of rational and pretzel tangle and knot groups into the affine group $\mathrm{AGL}(1,\mathbb{C})$, via a TQFT that is valued in the category of spans of singular vector bundles over $\mathbb{C}^{\ast}$. For these families, we derive closed-form expressions for their Alexander polynomials and establish bounds on their zeros. Finally, we specialize the functor at $t=-1$ and analyze colorings of rational tangles in terms of spans of complex vector spaces.

Affine representations of rational and pretzel knots

TL;DR

The paper develops a TQFT-based framework that realizes rational and pretzel knot groups as -representations through spans of vector spaces, enabling explicit computations of Alexander invariants for these families. By constructing and analyzing the functors and on tangles, the authors derive closed-form Alexander polynomials expressed via continuants and quantum integers, and establish zeros-location results (e.g., unit-circle zeros for many pretzel cases). A detailed treatment of 2-bridge knots shows how Alexander polynomials arise as equalizers of tangle-maps, while pretzel knots are handled through a block-structured, cyclic tridiagonal framework yielding determinant formulas. The specialization connects to rational-tangle colorings, producing a geometric description of rational tangles as a rational curve in and linking coloring fractions to Grassmannian slopes. Overall, the approach unifies representation-theoretic, combinatorial, and geometric perspectives on Alexander-type invariants for a broad class of knots and tangles with concrete, calculable expressions.

Abstract

We construct and study representations of rational and pretzel tangle and knot groups into the affine group , via a TQFT that is valued in the category of spans of singular vector bundles over . For these families, we derive closed-form expressions for their Alexander polynomials and establish bounds on their zeros. Finally, we specialize the functor at and analyze colorings of rational tangles in terms of spans of complex vector spaces.

Paper Structure

This paper contains 12 sections, 47 theorems, 144 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. ${\textup{AGL}_{1}}(\mathbb{C})$-representations and Alexander invariants
  3. Quantum integers
  4. 2-tangles
  5. 2-bridge knots
  6. Pretzel knots
  7. Odd pretzel knots $K=P(2k_1+1,\ldots, 2k_{n}+1)$
  8. Even pretzel knots $K(2k_1,2k_2+1,\ldots,2k_n+1)$
  9. $n=2p$
  10. $n=2p+1$
  11. The functor ${\mathcal{A}}_{-1}$
  12. Colorings

Key Result

Theorem 1

There exists a braided monoidal functor computing the Alexander module of knots.

Figures (8)

  • Figure 2: Orientations for the generators of the tangle category
  • Figure 3: Operations for 2-tangles
  • Figure 4: The tangle $R(T)$
  • Figure 5: Codirected and alternating braidings
  • Figure 6: Closures of a rational tangle in 3-strand form
  • ...and 3 more figures

Theorems & Definitions (92)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Conjecture : Hoste
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Remark 2.1
  • ...and 82 more