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Random colorings in manifolds

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Abstract

We develop a general method for constructing random manifolds and sub-manifolds 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.

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Authors and Affiliations

  1. Department of Mathematics, Technion—Israel Institute of Technology, Haifa, 32000, Israel

    Chaim Even-Zohar

  2. Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, CA, 95616, USA

    Joel Hass

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  1. Chaim Even-Zohar
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  2. Joel Hass
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Correspondence to Chaim Even-Zohar.

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Dedicated to Nati Linial, our friend, colleague and teacher

This work was carried out in part while the first author was at the Alan Turing Institute, supported by Lloyds Register Foundation’s programme on Data-Centric Engineering. This work was carried out while the second author was visiting the Oxford Mathematical Institute and ISTA, and was a Christensen Fellow at St. Catherine’s College. The second author was partially supported by NSF grant DMS:FRG 1760485 and BSF grant 2018313. Most of the computational work was performed with the facilities of the School of Computer Science and Engineering at the Hebrew University of Jerusalem.

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Even-Zohar, C., Hass, J. Random colorings in manifolds. Isr. J. Math. 256, 153–211 (2023). https://doi.org/10.1007/s11856-023-2509-5

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