Ropelength and Writhe Quantization of 12-Crossing Knots

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 correlated with the signature and Rasmussen s invariant of the knots.

Keywords:

• knot
• topology
• ropelength

Acknowledgments

We thank Andrzej Stasiak and Radmilla Sazdanovic for helpful discussions, and Allison Moore, Charles Livingston, and Se-Goo Kim for providing the knot coordinates. We are generally grateful to everyone who develops, maintains, and answers emails about freely available tools for knot theory investigations, including Rob Scharein, Luca Tubiana, Dror Bar-Natan, Jason Cantarella, Brian Gilbert and others.

Notes

1 Specifically, T(9,8) has an Alexander value of 9 and a ropelength of 260.0, T(21,4) has an Alexander of 21 and a ropelength of of 275.5, and T(63,2) has an Alexander of–63 and a ropelength likely between 486 and 532 [18]

Additional information

Funding

This work is supported by the National Science Foundation, grant number 2105113.