There is 177,147 different tie knots according to this paper.
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.