Knots in interaction

TL;DR

This paper investigates the geometric interactions of knotted solitons, revealing how they can split, join, and form links while preserving total linking number, with implications for understanding complex topological structures.

Contribution

It introduces a detailed geometric framework for knotted soliton interactions, including local processes and the role of four-point vertices and duality transformations.

Findings

Interactions are local and can be three- or four-body processes.

Total linking number is conserved during interactions.

Solitons can twist, coil, and form links through variable twist and writhe.

Abstract

We study the geometry of interacting knotted solitons. The interaction is local and advances either as a three-body or as a four-body process, depending on the relative orientation and a degeneracy of the solitons involved. The splitting and adjoining is governed by a four-point vertex in combination with duality transformations. The total linking number is preserved during the interaction. It receives contributions both from the twist and the writhe, which are variable. Therefore solitons can twine and coil and links can be formed.