Turing’s Knots

Alan Turing’s Knots

Alan Turing proposed the following definition of a knot.

A letter is one of A, B, C, D, E, F. A word is a sequence of letters, written horizontally. A word is balanced iff there are as many A’s as there are B’s; as many C’s as D’s, and as many E’s as F’s. A word is a knot iff it is balanced and no proper contiguous subsequence of it is balanced.

One might pause here are try to see how this is connected to the conventional topological notion of knot.

Two knots are equivalent if some sequence of the following steps, or their reverse, leads from one knot to the other. It is required that the word be a knot at each step:

Replace each letter the word by its double.
Delete some A and some B, or some C and some D, or some E and some F.

This definition is much easer than the topological one and highly suitable to computers.

Spoiler

Here is a program that plots the knot from the word. This tests a word for being a knot.

ACCCBBDDEAAACFFBBEEEDDFF is a trefoil knot.

Chiral and Reversible knots.