Random Colorings in Manifolds

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.

Figures (9)

• Figure 1: Examples of 2-simplices colored with one, two and three colors. The 2-color classes are the arcs where two colors meet, and the center point of the rightmost triangle is a 3-color class. Note that the center point always has the colors of all vertices in the simplex.
• Figure 2: The braid triangulation of the cube $[0,1]^3$. The 3-simplices correspond to all $3!=6$ orderings $x_i \leq x_j \leq x_k$ of the coordinates $x_1,x_2,x_3$. They all meet at the diagonal edge $x_1=x_2=x_3$, and some pairs intersect at 2-simplices on the planes $x_i = x_j$. On cubical 2-faces, they induce $\boxslash$, the braid triangulation of the 2-cube.
• Figure 3: A 2-coloring of a 2-simplex gives a 2-color class that is a 1-manifold. The intersection of the blue and red regions in a square is a 1-complex that is not a 1-manifold at the central vertex, where four blue-red arcs meet at a point.
• Figure 4: On the left the Voronoi 2-coloring of a 2-simplex. On the right, its linear 2-coloring $c^{\star}$.
• Figure 5: The $(d+1)$-colored triangulated prism $\sigma \times [0,1]$ in dimensions $d=1,2,3$ as in the proof of Lemma \ref{['double']}. Each colored region is a union of smaller simplices obtained by barycentric subdivision, but this is not shown. An isotopy takes each $m$-color class at height 0 to an $(m+1)$-color class at height $\tfrac{1}{2}$.

Theorems & Definitions (55)

• Definition 1
• Definition 2
• Definition 3
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof