As part of the tie knot generator, Michael Vejdemo-Johansson gives an algorithm for selecting a random knot from his enumeration and a random subset of the available locations where a U could be inserted. It shows a sequence of knotting cartoons that demonstrate how to tie that specific knot. In this section, every tie knot crosses the active blade over or under the other blade. The first instruction presupposes that you have done so in a way that places the seam of the tie in the location indicated in the illustration above.
Using the information you’ve learned, you can learn more about tie knot grammar and try to piece together a tie knot of your own. Interactive explorers for Fink and Mao’s original language and the Singly Tucked tie-knot language, which contains more recent and sophisticated knots, have been developed by the researchers. For the Full tie-knot language, they have a third interactive explorer available.
You may find information about the grammars we provided on their grammar page, including their respective knot counts and generating functions.