# Tropically constructed Lagrangians in mirror quintic threefolds

## TL;DR

This paper constructs numerous Lagrangian rational homology spheres in mirror quintic threefolds using tropical geometry and toric degeneration, revealing a rich family of non-Hamiltonian isotopic Lagrangians with weights linked to tropical curve multiplicities.

## Contribution

It introduces a novel method to construct Lagrangians in Calabi-Yau threefolds via tropical curves and toric degenerations, connecting tropical geometry with symplectic topology.

## Key findings

- Constructed over 300 disjoint Lagrangians in an example
- Established a correspondence between Lagrangian weights and tropical curve multiplicities
- Demonstrated the existence of many non-Hamiltonian isotopic Lagrangians

## Abstract

We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians. We check in an example that $>300$ mutually disjoint curves (and hence Lagrangians) arise. We show that the weight of each of these Lagrangians equals to the multiplicity of the corresponding tropical curve.