Knotting fingerprints resolve knot complexity and knotting pathways in ideal knots

Abstract

We use disk matrices to define knotting fingerprints that provide fine-grained insights into the local knotting structure of ideal knots. These knots have been found to have spatial properties that highly correlate with those of interesting macromolecules. 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 as the knot is trimmed progressively to a short arc unknot. A specialization of the Cheeger constant provides a measure of constraint on these independent unknotting pathways. Furthermore, the structure of the knotting fingerprint supports a comparison of the tight knot pathways to the unconstrained unknotting pathways of comparable length.