The Jones polynomial of collections of open curves in 3-space

TL;DR

This paper introduces a new definition of the Jones polynomial for collections of open curves in 3-space, enabling topological and geometrical complexity analysis of such collections, relevant to physical systems like polymers.

Contribution

It generalizes the Jones polynomial to open curves and linkoids, providing a topological invariant that is continuous and applicable to complex open curve collections.

Findings

The Jones polynomial for open curves is well-defined and coincides with that of the corresponding link when endpoints meet.

It is a continuous function of curve coordinates with real coefficients.

Numerical examples demonstrate its effectiveness in characterizing topological complexity.

Abstract

Measuring the entanglement complexity of collections of open curves in 3-space has been an intractable, yet pressing mathematical problem, relevant to a plethora of physical systems, such as in polymers and biopolymers. In this manuscript, we give a novel definition of the Jones polynomial that generalizes the classic Jones polynomial to collections of open curves in 3-space. More precisely, first we provide a novel definition of the Jones polynomial of linkoids (open link diagrams) and show that this is a well-defined single variable polynomial that is a topological invariant, which, for link-type linkoids, it coincides with that of the corresponding link. Using the framework introduced in Panagiotou and Kauffman 2020 arXiv:2001.01303, this enables us to define the Jones polynomial of collections of open and closed curves in 3-space. For collections of open curves in 3-space, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates. As the endpoints of the curves tend to coincide, the Jones polynomial tends to that of the resultant link. We demonstrate with numerical examples that the novel Jones polynomial enables us to characterize the topological and geometrical complexity of collections of open curves in 3-space for the first time.