Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Signs of knot polynomial evaluations from a topological perspective

Luana Jost, Lukas Lewark

TL;DR

This work connects topological data from the double branched cover $D_K$ of a knot to precise evaluations of the Jones and $Q$-polynomials at special roots of unity. It develops singular determinants $\delta_p$ and a linking-form framework to express these evaluations in terms of $H_1(D_K;\mathbb{Z})$ and its $p$-primary structure, providing explicit formulae that involve the dimension of $H_1(D_K;\mathbb{F}_p)$ and invariants $r_{p,k}$. A key contribution is identifying the Lipson invariant $\varepsilon_K$ with $\delta_3(K)$ and extending to $\delta_5$ for the $Q$-polynomial, enabling new unknotting-number obstructions and reinforcing the topological meaning of these quantum invariants. The paper also clarifies the limits of extending these determinations from knots to links by presenting concrete counterexamples, thereby delimiting the scope of a purely 3-manifold–invariant interpretation of these polynomial evaluations.

Abstract

We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of the double branched covering. We provide explicit formulae for these evaluations in terms of the linking pairing. The proof proceeds via so-called singular determinants, from which we also extract new lower bounds for the unknotting numbers of knots and links.

Signs of knot polynomial evaluations from a topological perspective

TL;DR

This work connects topological data from the double branched cover of a knot to precise evaluations of the Jones and -polynomials at special roots of unity. It develops singular determinants and a linking-form framework to express these evaluations in terms of and its -primary structure, providing explicit formulae that involve the dimension of and invariants . A key contribution is identifying the Lipson invariant with and extending to for the -polynomial, enabling new unknotting-number obstructions and reinforcing the topological meaning of these quantum invariants. The paper also clarifies the limits of extending these determinations from knots to links by presenting concrete counterexamples, thereby delimiting the scope of a purely 3-manifold–invariant interpretation of these polynomial evaluations.

Abstract

We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the -polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of the double branched covering. We provide explicit formulae for these evaluations in terms of the linking pairing. The proof proceeds via so-called singular determinants, from which we also extract new lower bounds for the unknotting numbers of knots and links.
Paper Structure (5 sections, 15 theorems, 77 equations, 4 figures)

This paper contains 5 sections, 15 theorems, 77 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Singular link determinants
  3. Linking forms
  4. Evaluations of knot polynomials
  5. The case of links

Key Result

Theorem 1.0

Let a knot $K$ be given. Denote by $\alpha \geq 0$ and $q \geq 1$ the unique integers such that $3$ does not divide $q$ and $\det(K) = 3^\alpha q$. Then where for all $\eta \in \mathbb{Z}$ not divisible by $2$ or $3$,

Figures (4)

  • Figure 1: Seifert surfaces for $L_+$ and $L_-$.
  • Figure 2: The links appearing in \ref{['ex:counterex']}.
  • Figure :
  • Figure :

Theorems & Definitions (37)

  • Theorem 1.0
  • Theorem 1.0
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Theorem 2.6
  • ...and 27 more