no. 1
A quantum obstruction for purely cosmetic surgeries
Detcherry, Renaud1
1Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, Université Bourgogne Europe, 21000 Dijon, France
Annales de l'Institut Fourier, Volume 76 (2026) no. 1, pp. 229-247

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 $\pm \frac{1}{5k}$, except if $\smash{J_K\bigl (e^{\frac{2i\pi }{5}}\bigr )=1}$, where $J_K$ is the Jones polynomial of $K$. For any odd prime $r\ge 5$, we also give an obstruction for $K$ to have a $\pm \frac{1}{k}$ surgery slope with $r\nmid k$ 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 $\pm \frac{1}{5k}$, sauf si $\smash{J_K\bigl (e^{\frac{2i\pi }{5}}\bigr )=1}$, où $J_K$ est le polynôme de Jones de $K$. Pour tout nombre premier $r\ge 5$, nous donnons aussi une obstruction pour ce que $K$ ait une chirurgie purement cosmétique de pentes $\pm \frac{1}{k}$ avec $r\nmid k$ 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:
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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

Detcherry, Renaud  1

1 Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, Université Bourgogne Europe, 21000 Dijon, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Detcherry, Renaud. A quantum obstruction for purely cosmetic surgeries. Annales de l'Institut Fourier, Volume 76 (2026) no. 1, pp. 229-247. doi: 10.5802/aif.3673

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