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Skeins on Branes

Tobias Ekholm, Vivek Shende

TL;DR

This work establishes that HOMFLYPT skein relations emerge naturally from deformation-invariant counts of holomorphic curves with Lagrangian boundary in Calabi–Yau 3-folds, by organizing wall-crossings in moduli spaces into skein moves. It defines skein-valued curve counts taking values in the skein module of the Lagrangian and proves their invariance under a broad axiomatic perturbation framework, unifying open-string physics with Chern–Simons theory. The authors prove the Ooguri–Vafa conjecture by showing that the HOMFLYPT polynomial coefficients of a link in $S^3$ count holomorphic curves in the resolved conifold with boundary on a pushoff of the link conormal, using conifold transition and SFT stretching. The construction provides a rigorous mathematical realization of Witten’s open-string/Chern–Simons correspondence and offers a robust enumerative machinery for open Gromov–Witten theory in CY threefolds. Overall, the paper delivers a deformation-invariant, skein-valued counting framework that connects enumerative geometry, symplectic field theory, and quantum topology via precise wall-crossing analyses and knot-theoretic invariants.

Abstract

We study 1-parameter families of holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds. We show that the expected codimension-1 phenomena can be organized to match the HOMFLYPT skein relations from quantum topology. It follows that counting holomorphic curves by the class of their boundaries in the skein module of the Lagrangian gives a deformation invariant result. This is a mathematically rigorous incarnation of Witten's assertion that boundaries of open topological strings create line defects in Chern-Simons theory. Using this theory, we rigorously establish the following prediction of Ooguri and Vafa: the coefficients of the HOMFLYPT polynomial of a link in the three-sphere count the holomorphic curves in the resolved conifold, with boundary on (a pushoff of) the link conormal.

Skeins on Branes

TL;DR

This work establishes that HOMFLYPT skein relations emerge naturally from deformation-invariant counts of holomorphic curves with Lagrangian boundary in Calabi–Yau 3-folds, by organizing wall-crossings in moduli spaces into skein moves. It defines skein-valued curve counts taking values in the skein module of the Lagrangian and proves their invariance under a broad axiomatic perturbation framework, unifying open-string physics with Chern–Simons theory. The authors prove the Ooguri–Vafa conjecture by showing that the HOMFLYPT polynomial coefficients of a link in count holomorphic curves in the resolved conifold with boundary on a pushoff of the link conormal, using conifold transition and SFT stretching. The construction provides a rigorous mathematical realization of Witten’s open-string/Chern–Simons correspondence and offers a robust enumerative machinery for open Gromov–Witten theory in CY threefolds. Overall, the paper delivers a deformation-invariant, skein-valued counting framework that connects enumerative geometry, symplectic field theory, and quantum topology via precise wall-crossing analyses and knot-theoretic invariants.

Abstract

We study 1-parameter families of holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds. We show that the expected codimension-1 phenomena can be organized to match the HOMFLYPT skein relations from quantum topology. It follows that counting holomorphic curves by the class of their boundaries in the skein module of the Lagrangian gives a deformation invariant result. This is a mathematically rigorous incarnation of Witten's assertion that boundaries of open topological strings create line defects in Chern-Simons theory. Using this theory, we rigorously establish the following prediction of Ooguri and Vafa: the coefficients of the HOMFLYPT polynomial of a link in the three-sphere count the holomorphic curves in the resolved conifold, with boundary on (a pushoff of) the link conormal.

Paper Structure

This paper contains 29 sections, 26 theorems, 47 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Skein relations from moduli of holomorphic maps
  3. Skein-valued curve counting
  4. The Ooguri-Vafa conjecture
  5. Linking numbers of open curves and Lagrangians
  6. Local models for crossings and nodal curves
  7. Hyperbolic boundary
  8. Elliptic boundary
  9. Locally standard 1-parameter families
  10. Axiomatics
  11. Skein valued curve counts
  12. Definition
  13. Invariance
  14. Closed and open invariants
  15. Adequate perturbations in the absence of multiple covers
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let ${\mathcal{M}}$ be the moduli of bare disconnected $J$-holomorphic curves in $X$ such that each component has boundary and the total boundary is $\beta$. Then ${\mathcal{M}}$ is a transversely cut out compact oriented 0-manifold, and the HOMFLYPT polynomial $H_{K}(a,z)$ of $K$ is given by the fo where $(-1)^{o(u)}$ is the orientation sign of $[u] \in {\mathcal{M}}$.

Figures (4)

  • Figure 1: The framed HOMFLYPT skein relations.
  • Figure 2: Moduli spaces near nodal curves, top hyperbolic and bottom elliptic. Solid blue lines indicates parameterized moduli spaces of curves, with the interesting parts of the curves depicted above them. The green dashed line indicates the association of points in distinct moduli spaces that leads to the skein relation.
  • Figure 3: Relation between hyperbolic and elliptic nodes near a cusp.
  • Figure 4: Top picture shows the unique annulus with boundary on $L_{K}$ and along $K\subset S^{3}$, before SFT-stretching. As we SFT-stretch, the boundary on $S^{3}$ undergoes skein moves until, as illustrated in the bottom picture, all curves left an $\epsilon$-neighborhood of $S^{3}$.

Theorems & Definitions (77)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 67 more