Knots and Links from Random Projections

TL;DR

This paper investigates models of random knots and links generated by projecting fixed space curves in higher dimensions onto random 3D subspaces, analyzing curvature and linking number expectations.

Contribution

It introduces a framework for studying random knots and links via orthogonal projections from higher-dimensional spaces, providing methods to compute curvature and linking number moments.

Findings

Expectation of curvature varies with initial space curve.

Methods to compute second moment of linking number.

Models of random knots and links can be recovered by choosing initial curves.

Abstract

In this paper we study a model of random knots obtained by fixing a space curve in $n$-dimensional Euclidean space with $n>3$, and orthogonally projecting the space curve on to random $3$ dimensional subspaces. By varying the space curve we obtain different models of random parametrized knots, and we will study how the expectation value of the curvature changes as a function of the initial parametrized space curve. In the case when the initial data is a pair of space curves, or more generally a pair of manifolds satisfying certain conditions on their dimension, then we obtain models of random links for which we will give methods to compute the second moment of the linking number. As an application of our computations, we will study numerous models of random knots and links, and how to recover these models by appropriately choosing the initial space curves to be projected.