The bracket polynomial of bonded knots and applications to entangles proteins

TL;DR

This paper introduces a new mathematical invariant for bonded knots, modeling proteins with disulfide bonds, by extending the Kauffman bracket polynomial to account for intramolecular bonds, with potential applications in understanding protein entanglements.

Contribution

It develops a bonded version of the Kauffman bracket polynomial and proves its properties as an invariant for bonded knots, expanding knot theory tools for biological applications.

Findings

Bonded Kauffman bracket skein module is infinitely generated and torsion-free.

The invariant distinguishes bonded knots with intramolecular bonds.

Application potential in analyzing protein entanglements.

Abstract

We model proteins with intramolecular bonds, such as disulfide bridges, using the concept of bonded knots -- closed loops in three-dimensional space equipped with additional bonds that connect different segments of the knot. We extend the Kauffman bracket polynomial (which is closely related to the Jones polynomial) to bonded knots through the introduction of the bonded version of the Kauffman bracket skein module. We show that this module is infinitely generated and torsion-free for both the rigid and topological case of bonded knots, providing an invariant of such structures.