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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.
Diagram of how to tie the hitching tie knot. The hitching tie is a simple knot used to tie off stuff sacks that allows quick access as it unties quickly. [1] [2] To untie the knot, just pull hard on the free end of the rope and the knot will fall open. This is simply a noose or slip knot, with the loop tightened around an object.
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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 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
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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 ...