Random Colorings in Manifolds

TL;DR

This paper introduces a method for constructing and analyzing random submanifolds in arbitrary dimensions via vertex coloring of triangulated manifolds, with applications to knot theory and topological problem solving.

Contribution

It develops a general framework for random manifold generation, determines conditions for submanifold emergence, and applies these to study knots, links, and topological properties.

Findings

Probability of unknot decays exponentially in 3-colorings of 3-balls

Expected Euler characteristic computed for random submanifolds

Computer experiments find low-genus surfaces with prescribed boundary knots

Abstract

We develop a general method for constructing random manifolds and submanifolds in arbitrary dimensions. The method is based on associating colors to the vertices of a triangulated manifold, as in recent work for curves in 3-dimensional space by Sheffield and Yadin (2014). We determine conditions on which submanifolds can arise, in terms of Stiefel-Whitney classes and other properties. We then consider the random submanifolds that arise from randomly coloring the vertices. Since this model generates submanifolds, it allows for studying properties and using tools that are not available in processes that produce general random subcomplexes. The case of 3 colors in a triangulated 3-ball gives rise to random knots and links. In this setting, we answer a question raised by de Crouy-Chanel and Simon (2019), showing that the probability of generating an unknot decays exponentially. In the general case of k colors in d-dimensional manifolds, we investigate the random submanifolds of different codimensions, as the number of vertices in the triangulation grows. We compute the expected Euler characteristic, and discuss relations to homological percolation and other topological properties. Finally, we explore a method to search for solutions to topological problems by generating random submanifolds. We describe computer experiments that search for a low-genus surface in the 4-dimensional ball whose boundary is a given knot in the 3-dimensional sphere.