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Twisted Knots and the Perturbed Alexander Invariant

Joe Boninger

TL;DR

This paper investigates how the perturbed Alexander invariant $\rho_1$ behaves under coherent twisting of knots, proving that the coefficients grow linearly with the twist parameter and that the associated growth rate $d_t(\mathcal{K})$ converges to a rational function in $T$, with $(\lim_{t\to\infty} d_t)|_{T=1}=\pm\frac{n-1}{2n}$. It develops two robust tools—a random-walk model on tangle diagrams (via Green’s functions) and the Burau representation of full twists—to analyze twisting systematically, including a contraction operation on Markov chains that preserves key invariants. The paper also establishes stabilization of the Alexander polynomial under twisting and derives an explicit asymptotic form for $\rho_1$ in the limit of infinite twists, confirming that $\rho_1$ distinguishes infinitely many knots in twisted families (e.g., the $(2,2t+1)$ torus knots via $\rho_1 \to -1/(1+T)^2$). Beyond results on $\rho_1$, the work introduces a symmetric perturbed Conway invariant $\delta_1$ and conjectures about positivity obstructions, linking classical and quantum knot invariants and suggesting broader applicability of the random-walk and contraction framework. Overall, the study provides a concrete, computable description of $\rho_1$-growth under twisting and supplies new techniques with potential utility for higher-order perturbed invariants and positivity questions in knot theory.

Abstract

The perturbed Alexander invariant $ρ_1$, defined by Bar-Natan and van der Veen, is a powerful, easily computable polynomial knot invariant with deep connections to the Alexander and colored Jones polynomials. We study the behavior of $ρ_1$ for families of knots $\{K_t\}$ given by performing $t$ full twists on a set of coherently oriented strands in a knot $K_0 \subset S^3$. We prove that as $t \to \infty$ the coefficients of $ρ_1$ grow asymptotically linearly, and we show how to compute this growth rate for any such family. As an application we give the first theorem on the ability of $ρ_1$ to distinguish knots in infinite families, and we conjecture that $ρ_1$ obstructs knot positivity via a "perturbed Conway invariant." Along the way we expand on a model of random walks on knot diagrams defined by Lin, Tian and Wang.

Twisted Knots and the Perturbed Alexander Invariant

TL;DR

This paper investigates how the perturbed Alexander invariant behaves under coherent twisting of knots, proving that the coefficients grow linearly with the twist parameter and that the associated growth rate converges to a rational function in , with . It develops two robust tools—a random-walk model on tangle diagrams (via Green’s functions) and the Burau representation of full twists—to analyze twisting systematically, including a contraction operation on Markov chains that preserves key invariants. The paper also establishes stabilization of the Alexander polynomial under twisting and derives an explicit asymptotic form for in the limit of infinite twists, confirming that distinguishes infinitely many knots in twisted families (e.g., the torus knots via ). Beyond results on , the work introduces a symmetric perturbed Conway invariant and conjectures about positivity obstructions, linking classical and quantum knot invariants and suggesting broader applicability of the random-walk and contraction framework. Overall, the study provides a concrete, computable description of -growth under twisting and supplies new techniques with potential utility for higher-order perturbed invariants and positivity questions in knot theory.

Abstract

The perturbed Alexander invariant , defined by Bar-Natan and van der Veen, is a powerful, easily computable polynomial knot invariant with deep connections to the Alexander and colored Jones polynomials. We study the behavior of for families of knots given by performing full twists on a set of coherently oriented strands in a knot . We prove that as the coefficients of grow asymptotically linearly, and we show how to compute this growth rate for any such family. As an application we give the first theorem on the ability of to distinguish knots in infinite families, and we conjecture that obstructs knot positivity via a "perturbed Conway invariant." Along the way we expand on a model of random walks on knot diagrams defined by Lin, Tian and Wang.
Paper Structure (17 sections, 27 theorems, 175 equations, 7 figures)

This paper contains 17 sections, 27 theorems, 175 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Random walks, and full twists on braids
  3. A perturbed Conway invariant
  4. Further Discussion
  5. Outline
  6. Acknowledgements
  7. Background and conventions
  8. Upright diagrams and the $\rho_1$ invariant
  9. Random walks on tangle diagrams
  10. Simple cycles and the Alexander polynomial
  11. Contracting tangle Markov chains
  12. Random walks on powers of full twists
  13. Burau representation of powers of full twists
  14. Asymptotic behavior
  15. Stabilization of the Alexander polynomial
  16. ...and 2 more sections

Key Result

Theorem 1.1

For a family of knots $\mathcal{K} = \{K_t\}$ as above, twisted along $n$ coherently oriented parallel strands, define Then as $t \to \infty$, $d_t$ converges to a non-polynomial rational function in $T$. The limit, considered as a rational function, satisfies

Figures (7)

  • Figure 1: Performing a full twist on three coherently oriented strands
  • Figure 2: Crossing conventions
  • Figure 3: An upright long knot diagram of the unknot
  • Figure 4: $D_0$, $D_\infty$ and $D^\tau_\infty$ for the family of $(2,2t + 1)$-torus knots
  • Figure 5: A crossingless diagram with an infinite twist vertex
  • ...and 2 more figures

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Conjecture 1.5
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: ltw98
  • Lemma 2.4
  • proof
  • ...and 47 more