You've got your Windsor, your half-Windsor, and... well, that's pretty much it. Except, of course, for the 168,998 other ways that science has determined it's possible to knot a tie. That's a thousand times more than we previously thought. All it took to figure it out was a repeat viewing of The Matrix Reloaded.
Our laughably conservative previous knot estimate came from a 1999 study which claimed that only 85 tie configurations were possible. Which seems both reasonable and too much effort to double-check. Fortunately, mathematician Mikael Vejdemo-Johansson was so taken by the knot sported by the Merovingian (above) in the Matrix sequel—in fairness, it's the most interesting part of the movie—that he both sought out how to recreate it, and cross-referenced it with the list of known knots. It was not there.
As New Scientist explains, the original study at sorely estimated the determination of knot enthusiasts, and their willingness to make unorthodox tucks:
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.
And so Vejdemo-Johansson and his team went about deducing a more accurate number of knots. They arrived at 177,147, all of which you can try out at his random knot generator here. It's fun! You end up with mostly goofy constructions like this one:
It's a breakthrough for awkward fashion, but more importantly it's a reminder that questioning conventional wisdom—no matter how mundane—can only make us more informed citizens of the world. Which is maybe the most Matrix thing about this discovery of all. [New Scientist]