Algorithmic Symplectic Packing

TL;DR

This paper advances symplectic packing theory by developing an improved algorithm to compute packing widths for higher numbers of simplices in four and six dimensions, providing detailed optimal packing configurations.

Contribution

It introduces a modified algorithmic approach that extends the computation of simplex packing widths to larger values and identifies all optimal packing multisets.

Findings

Determined k-simplex packing widths for up to 13 simplices in dimension four.

Extended packing width calculations to 8 simplices in dimension six.

Identified all multisets enabling optimal packings.

Abstract

In this article we explore a symplectic packing problem where the targets and domains are $2n$-dimensional symplectic manifolds. We work in the context where the manifolds have first homology group equal to $\mathbb{Z}^n$, and we require the embeddings to induce isomorphisms between first homology groups. In this case, Maley, Mastrangeli and Traynor showed that the problem can be reduced to a combinatorial optimization problem, namely packing certain allowable simplices into a given standard simplex. They designed a computer program and presented computational results. In particular, they determined the simplex packing widths in dimension four for up to $k=12$ simplices, along with lower bounds for higher values of $k$. We present a modified algorithmic approach that allows us to determine the $k$-simplex packing widths for up to $k = 13$ simplices in dimension four and up to $k = 8$ simplices in dimension six. Moreover, our approach determines all simplex-multisets that allow for optimal packings.