The BiEntropy of Some Knots on the Simple Cubic Lattice

TL;DR

This paper investigates the complexity of knots on a cubic lattice by analyzing their binary representations and BiEntropy, revealing that more crossings and non-alternating knots tend to be more disordered.

Contribution

It introduces a method to quantify the disorder of lattice knots using BiEntropy and compares different knot types and complexities.

Findings

Binary encoded knots are highly disordered.

BiEntropy increases with knot complexity and length.

Non-alternating knots are more disordered than alternating ones.

Abstract

Binary representations of the trefoil and other knots of up to ten crossings in the simple cubic lattice were created. The BiEntropy of each knot was computed using a variety of binary encodings and compared against controls. This showed that binary encoded knots are highly disordered information objects. The BiEntropy of knots on the simple cubic lattice increases slightly as the number of crossings and length of encoding increases. We show that the non-alternating knots of nine and ten crossings are more disordered than the alternating knots of nine and ten crossings.