The Mathematics of Tie Knots

TL;DR

This paper classifies the knot types of 85 tie knots from a formal language model, showing they are all prime and alternating, and relates them to known mathematical knots.

Contribution

It provides a complete classification of Fink and Mao's tie knots within knot theory, identifying their properties and relation to known knot types.

Findings

All tie knots are prime knots.

All tie knots are alternating.

Tie knots include unknot, trefoil, twist, and torus knots.

Abstract

In 2000, Thomas Fink and Young Mao studied neck ties and, with certain assumptions, found 85 different ways to tie a neck tie. They gave a formal language which describes how a tie is made, giving a sequence of moves for each neck tie. The ends of a neck tie can be joined together, which gives a physical model of a mathematical knot that we call a tie knot. In this paper we classify the knot type of each of Fink and Mao's 85 tie knots. We describe how the unknot, left and right trefoil, twist knots and $(2,p)$ torus knots can be recognized from their sequence of moves. We also view tie knots as a family within the set of all knots. Among other results, we prove that any tie knot is prime and alternating.