A quantum obstruction for purely cosmetic surgeries

Detcherry, Renaud

doi:10.5802/aif.3673

Detcherry, Renaud ¹

¹ Institut de Mathématiques de Bourgogne, UMR 5584 CNRS Université Bourgogne Europe 21000 Dijon, France

Annales de l'Institut Fourier, Online first, 19 p.

Abstract
Résumé

We present new obstructions for a knot $K$ in $S^{3}$ to admit purely cosmetic surgeries, which arise from the study of Witten–Reshetikhin–Turaev invariants at fixed level, and can be framed in terms of the colored Jones polynomials of $K$ .

In particular, we show that if $K$ has purely cosmetic surgeries then the slopes of the surgery are of the form $\pm \frac{1}{5 k}$ , except if $J_{K} (e^{\frac{2 i π}{5}}) = 1$ , where $J_{K}$ is the Jones polynomial of $K$ . For any odd prime $r \geq 5$ , we also give an obstruction for $K$ to have a $\pm \frac{1}{k}$ surgery slope with $r ∤ k$ that involves the values of the first $\frac{r - 3}{2}$ colored Jones polynomials of $K$ at an $r$ -th root of unity. We verify the purely cosmetic surgery conjecture for all knots with at most $17$ crossings.

Nous donnons de nouvelles obstructions pour qu’un nœud $K$ dans $S^{3}$ admette des chirurgies purement cosmétiques, qui proviennent de l’étude des invariants de Witten-Reshetikhin-Turaev à niveau fixé.

En particulier, nous montrons que si $K$ a des chirurgies purement cosmétiques alors les pentes sont de la forme $\pm \frac{1}{5 k}$ , sauf si $J_{K} (e^{\frac{2 i π}{5}}) = 1$ , où $J_{K}$ est le polynôme de Jones de $K$ . Pour tout nombre premier $r \geq 5$ , nous donnons aussi une obstruction pour ce que $K$ ait une chirurgie purement cosmétique de pentes $\pm \frac{1}{k}$ avec $r ∤ k$ qui fait intervenir les premiers $\frac{r - 3}{2}$ polynômes de Jones coloriés de $K$ en une racine $r$ -ème de l’unité. Nous vérifions la conjecture pour tous les nœuds à moins de $17$ croisements.

Received: 2022-04-04
Revised: 2022-06-30
Accepted: 2022-09-23
Online First: 2025-02-10

DOI: 10.5802/aif.3673

Classification: 57M25
Keywords: Dehn surgery, cosmetic surgeries, Jones polynomial, Reshetikhin–Turaev TQFTs
Mots-clés : chirurgie de Dehn, chirurgies cosmétiques, polynôme de Jones, TQFTs de Reshetikhin-Turaev

Author's affiliations:

Detcherry, Renaud ¹

¹ Institut de Mathématiques de Bourgogne, UMR 5584 CNRS Université Bourgogne Europe 21000 Dijon, France

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Detcherry, Renaud. A quantum obstruction for purely cosmetic surgeries. Annales de l'Institut Fourier, Online first, 19 p.

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