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.