Knotting Probability of Equilateral Hexagons

2018/10/28 by Hake, Kathleen
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1810.11882

Abstract

For a positive integer n≥ 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≤ 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.

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