The Shapes of Tight Composite Knots

TL;DR

This paper computes tight shapes of numerous composite knots using a specialized algorithm, expanding the dataset and analyzing conjectures about their ropelengths and writhes, contributing to knot theory understanding.

Contribution

It provides the largest dataset of tight composite knots and tests two key conjectures about their geometric properties.

Findings

Ropelengths of composite knots are at least 4π-4 less than the sum of prime factors.

Writhes of composite knots approximate the sum of the prime factors' writhes.

Expanded dataset enables more comprehensive analysis of tight knot properties.

Abstract

We present new computations of tight shapes obtained using the constrained gradient descent code RIDGERUNNER for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of composite knots are at least 4\pi-4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors.