Skein recursion for holomorphic curves and invariants of the unknot

Tobias Ekholm; Vivek Shende

doi:10.5802/crmath.754

Research article - Geometry and Topology

Skein recursion for holomorphic curves and invariants of the unknot

Tobias Ekholm ¹; Vivek Shende ^2,³

¹Department of Mathematics and Centre for Geometry and Physics Uppsala University, Box 480, 751 06 Uppsala, Sweden and Institut Mittag-Leffler, Aurav 17, 182 60 Djursholm, Sweden
²Center for Quantum Mathematics, Syddansk Univ., Campusvej 55, 5230 Odense, Denmark
³Department of Mathematics, UC Berkeley, 970 Evans Hall, Berkeley CA 94720, USA

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1543-1554

Abstract (Original language)
Résumé

We determine the skein-valued Gromov–Witten partition functions for single toric Lagrangian branes in $C^3$ or in the resolved conifold. We first show geometrically they must satisfy certain skein-theoretic recursions, and then solve these equations. The recursion is a skein-valued quantization of the equation of the mirror curve. The solution is the expected hook-content formula.

Nous déterminons les fonctions de partition de Gromov–Witten à valeurs skein pour une brane lagrangienne torique dans $C^3$ ou dans sa résolution conifolde. Nous montrons d’abord géométriquement qu’elles doivent satisfaire une formule récursive et nous résolvons cette équation. La récurrence est une quantification à valeurs skein de l’équation de la courbe miroir. Le résultat est la formule attendue des équerres et des contenus pour les partitions.

Received: 2024-11-28
Revised: 2025-04-24
Accepted: 2025-04-25
Published online: 2025-12-12

DOI: 10.5802/crmath.754

Classification: 53D45, 53D42, 57K10
Keywords: Holomorphic curve, HOMFLYPT skein module, Lagrangian submanifold, multiple-cover formula
Mots-clés : Courbe holomorphe, module skein HOMFLYPT, sous-variété lagrangienne, formule de couverture multiple

Author's affiliations:

Tobias Ekholm ¹ ; Vivek Shende ^{2
,

3}

¹ Department of Mathematics and Centre for Geometry and Physics Uppsala University, Box 480, 751 06 Uppsala, Sweden and Institut Mittag-Leffler, Aurav 17, 182 60 Djursholm, Sweden
² Center for Quantum Mathematics, Syddansk Univ., Campusvej 55, 5230 Odense, Denmark
³ Department of Mathematics, UC Berkeley, 970 Evans Hall, Berkeley CA 94720, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Skein recursion for holomorphic curves and invariants of the unknot},
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     language = {en},
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Tobias Ekholm; Vivek Shende. Skein recursion for holomorphic curves and invariants of the unknot. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1543-1554. doi: 10.5802/crmath.754

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