Proteins fold into three-dimensional structures stabilized by intramolecular bridges — disulfide bonds that link distant parts of the chain. The topology of a closed protein chain with such bridges is not captured by classical knot theory: the bridges create additional constraints that standard knot invariants ignore. Bonded knots model this situation: knots with prescribed internal connections between strands.
This paper systematically classifies uncolored bonded knots with singularity number up to seven. The procedure generates all possible planar graphs, converts them into bonded knot diagrams, then uses the Yamada polynomial and brute-force Reidemeister moves to distinguish topological types.
The classification is the first complete tabulation for bonded knots in this range, following the century-old tradition of knot tables but in a richer structure. Each entry in the table corresponds to a topologically distinct way a protein backbone can thread through space while maintaining a specific pattern of intramolecular bonds. The table is finite: with seven or fewer bond crossings, there are a catalogueable number of distinct topologies. The Yamada polynomial — designed for spatial graphs, not just knots — provides the invariant strong enough to separate them.