Knot invariants and Calabi-Yau crystals

TL;DR

The paper establishes a precise link between Calabi-Yau crystal melts with non-compact branes and Chern-Simons knot invariants, showing that unknot and Hopf link generating functions arise naturally and can be resummed to the topological vertex. It demonstrates a detailed dictionary between crystal amplitudes and A-model vertex calculations across C^3, conifold, and two-wall geometries, and identifies the open brane free energy with a target-space Gopakumar-Vafa expansion. This provides a target-space interpretation of open Donaldson-Thomas invariants and suggests a compact, GV-like expression for brane amplitudes. The work also lays out open questions on extending to more knots, gluing crystals to general toric geometries, and representing higher-dimensional holonomy data within the crystal framework.

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.

Figures (8)

• Figure 1: Three Lagrangian branes inserted on the positive slice and two Lagrangian antibranes inserted in the negative slice of the toric geometry of the ${{\mathbb C}}^3$ crystal.
• Figure 2: The toric geometry of the conifold crystal ending in a wall at the $y$ axes at distance corresponding to the Kahler parameter $t$.
• Figure 3: The toric geometry of the crystal model of ${\mathbb P}^1\times {\mathbb P}^1$, with two walls ending at the distances corresponding corresponding to the Kahler parameters $t_1$ and $t_2$.
• Figure 4: The toric diagram of the conifold of Kahler parameter $t$, with a brane inserted at distance $D$ on the compact leg and an antibrane inserted on a non-compact leg.
• Figure 5: The toric diagram corresponding to a brane inserted in the geometry ${\mathbb P}^1\times{\mathbb P}^1$ (with Kahler parameters $t_1$ and $t_2$) in the right compact leg at distance $D$ from the middle point.