Knotting Probability of Equilateral Hexagons

TL;DR

This paper investigates the probability of knotting in equilateral hexagons embedded in three-dimensional space, using symplectic geometry to parametrize the space and derive bounds on knotting likelihood.

Contribution

It introduces a symplectic geometric approach to parametrize equilateral hexagons and establishes new bounds on their knotting probability.

Findings

Parametrization of equilateral hexagons using action-angle coordinates

New bounds on the knotting probability of equilateral hexagons

Application of symplectic geometry techniques to geometric knot theory

Abstract

For a positive integer $n\ge 3$, the collection of $n$-sided polygons embedded in $3$-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded $n$-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components correspond to equilateral knot types. When $n\le 5$, the space of equilateral knots is connected. Therefore, we examine the space of equilateral hexagons. Using techniques from symplectic geometry, we can parametrize the space of equilateral hexagons with a set of measure preserving action-angle coordinates. With this coordinate system, we provide new bounds on the knotting probability of equilateral hexagons.