Ropelength and writhe quantization of 12-crossing knots

TL;DR

This study measures the ropelength and analyzes writhe quantization of all 12-crossing knots, revealing patterns and correlations with topological invariants, extending previous work on knots with fewer crossings.

Contribution

It provides the first comprehensive ropelength data for 12-crossing knots and demonstrates writhe quantization patterns in both alternating and non-alternating knots.

Findings

Writhe quantization observed near multiples of 4/7 for alternating knots.

Non-alternating knots show writhe quantization near multiples of 4/3.

Ropelength correlates positively with hyperbolic volume and Rasmussen s invariant.

Abstract

The ropelength of a knot is the minimum length required to tie it. Computational upper bounds have previously been computed for every prime knot with up to 11 crossings. Here, we present ropelength measurements for the 2176 knots with 12 crossings, of which 1288 are alternating and 888 are non-alternating. We report on the distribution of ropelengths within and between crossing numbers, as well as the space writhe of the tight knot configurations. It was previously established that tight alternating knots have a ``quantized'' space writhe close to a multiple of 4/7. Our data supports this for 12-crossing alternating knots and we find that non-alternating knots also show evidence of writhe quantization, falling near integer or half-integer multiples of 4/3, depending on the parity of the crossing number. Finally, we examine correlations between geometric properties and topological invariants of tight knots, finding that the ropelength is positively correlated with hyperbolic volume and its correlates, and that the space writhe is positively correlated with the Rasmussen s invariant.