Classification of (uncolored) bonded knots and links

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.

Figures (11)

• Figure 1: Representative examples of entangled protein topologies. Top: ribbon representations of protein structures; bottom: corresponding backbones with intramolecular bonds. a) classical protein knot formed by the backbone chain, b) cystine motif where two bridges form a pierced disk, c) theta-curve formed by a single covalent bond connecting two segments of the backbone, d) bonded knot formed by multiple intramolecular bonds, and e) lasso protein topology, where a backbone segment pierces the minimal surface spanned by the backbone and a disulfide bond.
• Figure 2: Reidemesiter moves for 3-valent knotted graphs.
• Figure 3: Enumeration of arcs of the bonded knot $B(3,1)_1.$
• Figure 4: Connected sums of bonded knots. The operations $\#_1$, $\#_2$, and $\#_3$ correspond respectively to the order 1, order 2, and order 3 connected sums described by Moriuchi Moriuchi2009.
• Figure 5: Replacing a 4-degree vertex with two possible crossing types.