Knot Fingerprints Resolve Knot Complexity and Knotting Pathways in Tight Knots

Knot fingerprints provide a fine-grained resolution of the local knotting structure of tight knots. From this fine structure and an analysis of the associated planar graph, one can define a measure of knot complexity using the number of independent unknotting pathways from the global knot type to the short arc unknot. A specialization of the Cheeger constant provides a measure of constraint on these independent unknotting pathways. Furthermore, the structure of the knot fingerprint supports a comparison of the tight knot pathways to the unconstrained unknotting pathways of comparable length.