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The Haves and The Have Knots

Ties. Lots of ties. Mathematicians have revealed there are over 177,000 distinct ways to knot a neck tie – more than 1000 times the number that was previously thought. They got their inspiration from an unusual style featured in the film The Matrix Reloaded.  Even John Kammeier doesn't know how to tie all of these knots.  

As well as vastly broadening the mathematical definition of a neck tie knot, the discovery means there are now hundreds of thousands of new ties to choose from. High fashion many of the styles are not but they have gone down well with Moss Bros, a UK-based formal menswear store. "Traditionalists will no doubt dismiss these, but I reckon modern dandies will lap it up," says company spokesman Dave Shaw.
Mathematics and ties have had a long-standing relationship. In 1999, Thomas Fink and Yong Mao of the University of Cambridge published a mathematical language describing tie knots in the journal Nature.

Merovingian madness

The pair used an existing tool from logic – known as formal language theory – to express the basic rules of tying a neck tie as a series of symbols. This included things like the placement of the tie, the direction of the fold and the need to end in a final tuck. They used their tie language to show that only 85 knots were possible.
Now mathematician Mikael Vejdemo-Johansson of the KTH Royal Institute of Technology in Stockholm, Sweden, has vastly broadened the tie landscape.
He first became interested in the subject after a YouTube tutorial caught his eye. It showed you how to copy an exotic-looking knot worn by Matrix villain, The Merovingian.
He went on to learn of other relatively new, exotic tie knots with names like Eldredge (see image, bottom left) and Trinity (see image, dead centre), which have cult followings online. "I was hooked," he says. Curious to learn more, he looked up the Fink-Mao paper – and immediately noticed that fancier knots like these weren't included.

Rules of tuck

It turns out Fink and Mao had made two assumptions about tie knots that drastically reduced the number available. They assumed that you would only make a tuck – folding one end of the tie under the rest to complete the knot – at the end of a given tying sequence, and that all knots would be covered by a flat stretch of fabric. Those assumptions don't hold for the new set of knots, which can involve making multiple tucks midway through a sequence – and surfaces with many folds and edges.
Vejdemo-Johansson and colleagues set about rewriting their language so it would include these more elaborate ties. One thing they had to do was simplify the language. The existing language described the process of tying a knot as a sequence of motions between the left, centre and right of the chest, moving the tie either away or towards the chest. Vejdemo-Johansson's team realised they could just describe moves as windings either clockwise or anticlockwise around the passive end of the tie, plus a tuck move. This freed them up to include much more elaborate ties.
They also changed an important rule: the limit to how many winding moves you can make before your tie gets embarrassingly short. Fink and Mao placed the limit at 8 for classical ties, but Vejdemo-Johansson's team chose 11 instead, as that is how many the Eldredge knot needs. Counting up all the possible windings and tucks before you hit this limit gives a total of 177,147 different tie knots.

Abstract fashion

Vejdemo-Johansson has created a website that generates random knots according to his rules if you want to have a go, though you might not win any sartorial awards.
"I have tried 10 or 20 of them, and most of them to be quite honest look kind of awkward," he says. In future, he would like to be able to mathematically identify nice knots. "This is pure speculation, but it seems to be the fraction of really attractive tie knots goes down when we widen the scope."
Instead, Vejdemo-Johansson recommends the Eldredge or Trinity knot, which he personally wears on a daily basis. The first looks better with a plain patterned tie while the second benefits from stripes, he says, and both attract attention. "At least every week or so, someone spontaneously comments on my tie."
Shaw of Moss Bros warns that for most people, the new knots won't be suitable for all situations. "One needs to have the right occasion to sport something so abstract."
FYI: Dan Hirsch, Meredith L Patterson, Anders Sandberg, Mikael Vejdemo-Johansson, in their own words, because I know that you wanted to know... djk

"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 and Mao (2000). 
We show that the relaxed tie knot description language that comprehensively describes these extended tie knot classes is either context sensitive or context free. It has a sub-language that covers all the knots that inspired the work, and that is regular. From this regular sub-language we enumerate 177 147 distinct tie knots that seem tieable with a normal necktie. These are found through an enumeration of 2 046 winding patterns that can be varied by tucking the tie under itself at various points along the winding."

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