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## Definition of Tie Knots

A tie knot is initiated by bringing the wide (active) end to the left and either over or under the narrow (passive) end, forming the triagonal basis and dividing the space into right, centre and left (R,C,L) regions, as shown in Fig. 1 (this, and all other figures, are drawn in the frame of reference of a mirror image of the actual tie).

Once begun, the knot is continued by wrapping the active end around the triagonal basis, as indicated, for example, in Fig. 2 with the Four-in-Hand; this process may be considered a sequence of half turns from one region to another. The location and orientation of the active end are represented by one of the six states , where R,C and L indicate the region from which the active end emanates and and denote the direction of the active end as viewed from in front, viz., out of the page (shirt) and into the page (shirt), respectively.

Figure 1: The two ways of beginning a knot, and , as viewed in a mirror. Both give rise to the triagonal basis and divide the space into the three regions through which the wide (active) end can subsequently pass. For knots beginning with , the tie must begin inside out.

The notational elements , etc., initially introduced as states, may be considered moves in as much as each represents the half-turn necessary to place the active end into the corresponding state. This makes the successive moves , for instance, impossible, and implies that is the inverse of . It follows that no two consecutive moves may indicate the same region or direction.

To complete a tie knot, the active end must be wrapped over the front, i.e., either or , then underneath to the centre, , and finally through (denoted T) the front loop just made.

Figure 2: The Four-in-Hand, represented by the sequence .

We can now formally define a tie knot as a sequence of moves chosen from the moveset , initiated by or and terminating with the subsequence or . The complete sequence is constrained such that the move direction oscillate between and and no two consecutive move regions be identical. Using this notation, the ubiquitous Four-in-Hand may be represented .

Next: Aesthetic Tie Knots Up: Tie Knots and Random Previous: Introduction

Yong Mao
Sat Nov 7 16:03:57 GMT 1998