Knot invariants and Calabi-Yau crystals

TL;DR

This paper demonstrates how Calabi-Yau crystals can produce knot invariants and connect to topological string theory, providing a new perspective on the computation of these invariants through crystal models.

Contribution

It establishes a direct link between Calabi-Yau crystal models and Chern-Simons knot invariants, clarifies the relation to the topological vertex, and introduces crystal resummation techniques.

Findings

Crystal models generate specific knot invariants.

Crystal amplitudes relate to the topological vertex.

Free energy can be expressed via Gopakumar-Vafa expansion.

Abstract

We show that Calabi-Yau crystals generate certain Chern-Simons knot invariants, with Lagrangian brane insertions generating the unknot and Hopf link invariants. Further, we make the connection of the crystal brane amplitudes to the topological vertex formulation explicit and show that the crystal naturally resums the corresponding topological vertex amplitudes. We also discuss the conifold and double wall crystal model in this context. The results suggest that the free energy associated to the crystal brane amplitudes can be simply expressed as a target space Gopakumar-Vafa expansion.