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Colored HOMFLYPT counts holomorphic curves.

Tobias Ekholm1,2, Vivek Shende3,4

  • 1Department of Mathematics and Centre for Geometry and Physics, Uppsala University, Uppsala 751 06, Sweden.

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This study connects link invariants to holomorphic curve counts in a Calabi-Yau threefold. The findings generalize previous work, revealing a skein-valued formula for annulus surfaces.

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Area of Science:

  • Topology
  • String Theory
  • Algebraic Geometry

Background:

  • Lagrangian conormals of links in the three-sphere can be mapped to the resolved conifold, a noncompact Calabi-Yau threefold.
  • Previous work established a link between curve counts and HOMFLYPT invariants for the uncolored case.

Purpose of the Study:

  • To demonstrate that the count of holomorphic curves ending on a transplanted Lagrangian conormal in the resolved conifold corresponds to HOMFLYPT invariants of all link colorings.
  • To generalize the identification of curve counts with link invariants to colored links.

Main Methods:

  • Transplanting Lagrangian conormals to the resolved conifold.
  • Developing a skein-valued multiple cover formula for isolated embedded annuli.
  • Utilizing string theoretic predictions by Ooguri and Vafa.

Main Results:

  • The count of all holomorphic curves in the resolved conifold ending on the transplanted Lagrangian is equivalent to the collection of HOMFLYPT invariants for all link colorings.
  • This result generalizes the previously identified curve count for uncolored links.

Conclusions:

  • The study provides a comprehensive framework connecting knot theory (HOMFLYPT invariants) with geometric concepts (holomorphic curves in Calabi-Yau manifolds).
  • The developed skein-valued multiple cover formula is a key tool for understanding these connections.