Bonded braids and the Markov theorem

TL;DR

The paper extends classical knot theory to bonded knots, modeling intramolecular bonds by attaching bonds to a knot and treating them as edge-colored spatial graphs. It develops a bonded braid theory via the topological bonded braid monoid $M_n$ (and rigid version $RM_n$), defines closures to recover bonded knots, and establishes an Alexander-type theorem for bonded knots. It then constructs Burau-type representations for bonded braids, including a reduced bonded Burau representation, and analyzes their (non)faithfulness in low dimensions. Finally, it proves bonded analogues of the Markov theorem, characterizing when two bonded braids have ambient isotopic closures and adapting Morton's threading approach to the bonded setting. The results provide a rigorous algebraic framework for bonded knots with potential applications to topology-informed models of protein folding and entanglement.

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.

Figures (28)

• Figure 1: Left: 3D ribbon model of the FS2 toxin peptide (pdb 1tfs). Right: corresponding bonded knot diagram with a direct closure. The multiple disulfide bonds stabilize the compact, active conformation of the toxin.
• Figure 2: Reidemeister moves for bonded knots. Any part of the depicted arcs can be either a knot arc or a band arc.
• Figure 3: Rigid versions of the move V. It was shown in gabrovvsek2021invariant that the move $\mathrm{V_R}$ is a consiquence of the move $\mathrm{V_R'}$.
• Figure 4: Isolating bonds via VI-moves.
• Figure 5: Reidemeister moves for knots with isolated bonds. Note that moves I, II, and III cannot be applied to a strand that represents a bond.

Theorems & Definitions (28)

• Theorem 1.1: gabrovvsek2021invariantKauffman1989
• Definition 1.1
• Lemma 1.1
• Lemma 1.2
• proof
• Lemma 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4