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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.
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
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).
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7 1 knot, septafoil knot, (7,2)-torus knot - a prime knot with crossing number seven, which can be arranged as a {7/2} star polygon ; 7 4 knot, "endless knot" 8 18 knot, "carrick mat" 10 161 /10 162, known as the Perko pair; this was a single knot listed twice in Dale Rolfsen's knot table; the duplication was discovered by Kenneth Perko
Friction knots are held in place by the friction between the windings of line. Knotted-ends knots are held in place by the two ends of the line being knotted together. Stopping may be either a temporary whipping or seizing, the commonest variety consisting of a few round turns finished off with a reef knot .