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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 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).
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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One of the fundamental structures used to tie knots. Specifically, it is a U-form narrower than a bight. [17] A type of knot used to create a closed circle in a line. A loop is one of the fundamental structures used to tie knots. It is a full circle formed by passing the working end of a rope over itself.
Pine pattern collar in tatting. Tatting is a technique for handcrafting a particularly durable lace from a series of knots and loops. [1] Tatting can be used to make lace edging as well as doilies, collars, accessories such as earrings, necklaces, waist beads, and other decorative pieces.