Order-forcing in Neural Codes

TL;DR

This paper introduces order-forcing, a combinatorial tool to analyze convex neural codes, enabling the construction of non-convex examples and demonstrating the tightness of a known dimension bound across all dimensions.

Contribution

The paper develops order-forcing as a new method to identify regions in neural codes that must align along line segments, expanding the understanding of convexity in neural codes.

Findings

Constructed new non-convex neural codes using order-forcing.

Expanded existing families of neural code examples.

Proved the tightness of a dimension bound in all Euclidean dimensions.

Abstract

Convex neural codes are subsets of the Boolean lattice that record the intersection patterns of convex sets in Euclidean space. Much work in recent years has focused on finding combinatorial criteria on codes that can be used to classify whether or not a code is convex. In this paper we introduce order-forcing, a combinatorial tool which recognizes when certain regions in a realization of a code must appear along a line segment between other regions. We use order-forcing to construct novel examples of non-convex codes, and to expand existing families of examples. We also construct a family of codes which shows that a dimension bound of Cruz, Giusti, Itskov, and Kronholm (referred to as monotonicity of open convexity) is tight in all dimensions.