More ties than we thought

Abstract

We extend the existing enumeration of neck tie-knots to include tie-knots with a textured front, tied with the narrow end of a tie. These tie-knots have gained popularity in recent years, based on reconstructions of a costume detail from The Matrix Reloaded, and are explicitly ruled out in the enumeration by Fink & Mao (2000). We show that the relaxed tie-knot description language that comprehensively describes these extended tie-knot classes is context free. It has a regular sub-language that covers all the knots that originally inspired the work. From the full language, we enumerate 266,682 distinct tie-knots that seem tie-able with a normal neck-tie. Out of these 266,682, we also enumerate 24,882 tie-knots that belong to the regular sub-language.

Study of knots in material culture

“…It is interesting to note that whilst there are 85 possibilities, only four of these are commonly used as ways to tie a neck tie. Whilst Fink and Mao only considered neck ties tied with the wide end of the neck tie, Hirsch et al [15] extended the neck tie knot possibilities by including those tied with the thin end. This takes the number of possible neck ties, with up to 13 moves, up to a staggering 177,146.…”

The Mathematics of Tie Knots

“…These tie knots do not satisfy condition (3) above and so are not found in Fink and Mao's list. As discussed in the introduction, these ties, and many others are found in [5].…”

“…In 2015, Dan Hirsch, Ingemar Markström, Meredith L. Patterson, Anders Sandberg and Mikael Vejdemo-Johansson [5] extended Fink and Mao's enumeration of neck ties to include ties with a textured front, and those tied with the narrow end of a tie. They created a new formal language to do so, and counted 266, 682 tie-knots that seem tie-able.…”

“…Indeed, N. Scoville [14] has an interesting webpage where he shows many other interesting neck ties. Many of these do not appear to obey even the more relaxed conditions of [5]. (The Atomic knot, the Danish knot and the Lady slipper knot are a few examples showing extra twists and loops which may not be captured by the formal language of [5].…”