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

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.

References in corpus (4)

• Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity
• A study of the entanglement in systems with periodic boundary conditions
• Knot polynomials of open and closed curves
• Vassiliev measures of complexity for open and closed curves in 3-space

Cited by in corpus (3)

• The Jones polynomial in systems with Periodic Boundary Conditions
• The Virtual Spectrum of Linkoids and Open Curves in 3-space
• Self-assembly Monte Carlo reveals localized entanglement in giant polymer melts