tl;dr: A tie knot generation toy by Mikael Vejdemo-Johansson
In 2000, Cambridge physicists Fink and Mao figured out a way to list all possible tie knots. They did it by creating a formal language to describe tie knots. However, they limited their language to fit their idea of a tie knot: tied with the broad blade, and finished with a flat front.
In 2012, a series of youtube videos by Alex Krasny went viral online, with instructions to tie tie knots like the Trinity and the Eldredge. These knots are not in the enumeration by Fink and Mao; they don't have a flat front, by design.
During 2013, I have worked out, in collaboration with Anders Sandberg, Meredith L. Patterson and Dan Hirsh, the ramifications of removing Fink and Mao's restrictions. We have condensed the formal language proposed by Fink and Mao to a language with (almost) no axioms and three symbols: W, T, U.
T is a clockwise (turnwise) move of the knot-tying blade, W is a counter-clockwise move, and U tucks the blade under a previous bow. Whether to start with an inwards or outwards crossing can be deduced by counting the total number of W and T in the knot description string, and all possible strings in W and T produce possible tie knots.
For known and named tie knots, we have compiled a list of different notations and enumerations.
We have a preprint on these results up on the arXiv.
In the tie knot generator we provide a generator that picks out a random knot from our enumeration, and picks a random subset of the possible places where a U could be placed. It displays a sequence of knotting cartoons to show how to tie that particular knot. Every tie knot here starts with crossing the active blade over or under the other blade, and the first instruction assumes you have done this in a way that places the seam of the tie as indicated in that picture.
You can explore the tie knot grammars and try to piece together a tie knot of your own. We have built interactive explorers for the original language by Fink and Mao and for the Singly Tucked tie-knot language that includes more modern and more complicated knots. We have a third interactive explorer for the Full tie-knot language
You can to read the grammars we specified, and their corresponding knot counts and generating functions at our grammar page.
We have collected our press mentions on a separate page.