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The discovery of all possible ways to tie a tie depends on a mathematical formulation of the act of tying a tie. In their papers (which are technical) and book (which is for a lay audience, apart from an appendix), the authors show that necktie knots are equivalent to persistent random walks on a triangular lattice, with some constraints on how the walks begin and end.
A small knot. The small knot, also known as oriental knot, Kent knot, or simple knot, is the simplest method of tying a necktie. Unlike the Four-in-hand knot and Windsor knot, the small knot is not self-releasing. The small knot is tied inside out, though this can be mitigated by giving the tie a half-twist during the tying process.
The Pratt knot uses less length than the half-Windsor or Windsor knots, and so is well suited to shorter ties or taller men. Unlike the four-in-hand knot, the Pratt method produces a symmetrical knot. It is of medium thickness. Using notation from and according to The 85 Ways to Tie a Tie, the knot is tied Lo Ci Lo Ri Co T (knot 5).
Hitching tie – simple knot used to tie off drawstring bags that allows quick access; Honda knot a.k.a. lariat loop – loop knot commonly used in a lasso; Hoxton knot – a method of arranging a scarf about the neck; Hunter's bend a.k.a. rigger's bend – joins two lines
The four-in-hand knot is tied by placing the tie around the neck and crossing the broad end of the tie in front of the narrow end. The broad end is folded behind the narrow end and brought forward on the opposite side, passed across the front horizontally, folded behind the narrow end again, brought over the top of the knot from behind, tucked behind the horizontal pass, and the knot pulled snug.
The Grantchester knot is a self-releasing, asymmetric way of tying a necktie. Using the notation presented in The 85 Ways to Tie a Tie , it is a Lo Ri Lo Ri Co Li, finishing with Ro Li Co T. Grantchester knot instructions
A Matthew Walker knot is tied in a circular bundle of any number of strands. To tie the knot, the tier takes each strand and forms a loop around the rest of the bundle, then passes the end through the newly formed loop to form an overhand knot. They then move to the next strand over, moving around the bundle in the direction they pass the loops.
Christensen knot (also "cross knot"): An elongated, symmetrical knot, whose main feature is the cruciform structure made by knotting the necktie through the double loop made in the front (Li Ro Ci Lo Ri Lo Ri Co T T, Knot 25 2). While it can be made with modern neckties, it is most effective with thinner ties of consistent width, which fell out ...