Bonded braids and the Markov theorem

TL;DR

This paper extends classical knot theory by developing algebraic tools for bonded knots, which model protein structures with disulfide bridges, and proves foundational theorems relating bonded braids to bonded knots.

Contribution

It introduces the bonded braid monoid, proves bonded analogues of the Alexander and Markov theorems, and extends braid representations to the bonded setting.

Findings

Bonded braid monoid encodes classical crossings and bonded connections.

Bonded Alexander and Markov theorems establish correspondence between bonded braids and bonded knots.

Extended Burau representations analyzed for faithfulness in low dimensions.

Abstract

Bonded knots arise naturally in topological protein modeling, where intramolecular interactions such as disulfide bridges stabilize folded configurations. These structures extend classical knot theory by incorporating embedded graphs, and have been formalized as bonded knots. In this paper, we develop the algebraic theory of bonded braids, introducing the bonded braid monoid in the topological and rigid settings, which encodes both classical braid crossings and (rigid) bonded connections. We prove bonded analogues of the Alexander and Markov theorems, establishing that every bonded knot arises as the closure of a bonded braid and that two bonded knots are equivalent if and only if their braid representatives are related by a finite sequence of algebraic (Markov-like) moves. In addition, we define the bonded Burau and reduced bonded Burau representations of the monoid, extending classical braid group representations to the bonded setting, and analyze their (non-)faithfulness in low dimensions.