Classification of (uncolored) bonded knots and links

TL;DR

This paper introduces a systematic method for classifying uncolored bonded knots with up to seven singularities, modeling protein chains with intramolecular bridges using topological and graph-based techniques.

Contribution

It develops a novel classification procedure combining graph generation, polynomial invariants, and Reidemeister moves for bonded knots.

Findings

Classified bonded knots with up to seven singularities.

Established a procedure for topological distinction of bonded knots.

Provided a catalog of bonded knot types for biological modeling.

Abstract

We present a systematic classification of uncolored bonded knots with singularity number at most seven. Bonded knots provide a topological model for closed protein chains with intramolecular bridges, such as disulfide bonds. Following the tradition of knot tabulation, we describe a procedure based on the generation of planar graphs, their conversion into bonded knot diagrams, and the use of the Yamada polynomial together with brute-force Reidemeister moves to distinguish topological knotted types.