An invariant for colored bonded knots

TL;DR

This paper introduces a new invariant called the HOMFLYPT skein module for colored bonded knots, modeling complex structures like protein chains with bridges, and analyzes its algebraic properties.

Contribution

It defines the HOMFLYPT skein module for colored bonded knots and characterizes its generators in rigid and non-rigid cases, extending knot invariants to complex bonded structures.

Findings

Rigid module is generated by colored Θ-curves and handcuff links.

Non-rigid module is generated by trivially embedded Θ-curves.

Non-rigid module does not detect knottedness of bonds.

Abstract

We equip a knot $K$ with a set of colored bonds, that is, colored intervals properly embedded into $\mathbb{R}^3 \setminus K$. Such a construction can be viewed as a structure that topologically models a closed protein chain including any type of bridges connecting the backbone residues. We introduce an invariant of such colored bonded knots that respects the HOMFLYPT relation, namely the HOMFLYPT skein module of colored bonded knots. We show that the rigid version of the module is freely generated by colored $\Theta$-curves and handcuff links, while the non-rigid version is freely generated by the trivially embedded $\Theta$-curve. The latter module, however, does not provide information about the knottedness of the bonds.