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TIE KNOTS AND RANDOM WALKS

Thomas Fink and Yong Mao

The simplest of conventional tie knots, the Four-in-Hand, has its origins in late nineteenth-century England. The Duke of Windsor, after abdicating in 1936, has been credited with introducing what is now known as the Windsor knot, whence its smaller derivative, the Half-Windsor, evolved. More recently, in 1989, the Pratt knot was revealed on the front page of the New York Times, the first new knot to appear in 50 years.

Rather than wait another half-century for the next sartorial advance, here we present a more formal approach. We introduce a mathematical model of tie knots and provide a map between tie knots and persistent random walks on a triangular lattice. We classify knots according to their size and shape and quantify the number of knots in each class. The optimal knot in a class is selected by the proposed aesthetic conditions of symmetry and balance. Of the 85 knots which can be tied with a conventional tie, we recover the four knots in widespread use and introduce six new aesthetic ones.

A tie knot is initiated by bringing the wide (active) end to the left and either over or under the narrow (passive) end, dividing the space into right, centre and left (R,C,L) regions (Fig. 1a). The knot is continued by subsequent half-turns, or moves, of the active end from one region to another (Fig. 1b) such that its direction alternates between out of the shirt and into the shirt ( ). To complete a knot, the active end must be wrapped from the right (left) over the front to the left (right), underneath to the centre and finally through (denoted T but not considered a move) the front loop just made.

Elements of the move set designate the moves necessary to place the active end into the corresponding region and direction. We can then define a tie knot as a sequence of moves initiated by or and terminating with the subsequence or . The sequence is constrained such that no two consecutive moves indicate the same region or direction.

We represent knot sequences as random walks on a triangular lattice (Fig. 1c). The axes r,c,l correspond to the three move regions R,C,L and the unit vectors represent the corresponding moves; we omit the directional notation and the terminal action T. Since all knot sequences end with and alternate between and , all knots of odd numbers of moves begin with while those of even numbers of moves begin with . Our simplified random walk notation is thus unique.

The size of a knot, and the primary parameter by which we classify it, is the number of moves in the knot sequence, denoted by the half-winding number h. The initial and terminal sequences dictate that the smallest knot be given by the sequence , with h=3. Practical (viz., the finite length of the tie) as well as aesthetic considerations suggest an upper bound on knot size; we limit our exact results to half-winding number .

The number of knots as a function of size, K(h), corresponds to the number of walks of length h beginning with and ending with or . It may be written

where K(1) = 0, and the total number of knots is .

The shape of a knot depends on the number of right, centre and left moves in the tie sequence. Since symmetry dictates an equal number of right and left moves (see below), knot shape is characterised by the number of centre moves . We use it to classify knots of equal size h; knots with identical h and belong to the same class. While a large centre fraction indicates a broad knot (e.g., the Windsor) and a small centre fraction suggests a narrow one (e.g., the Four-in-Hand), not all centre fractions allow aesthetic knots. We consequently limit our attention to .

The number of knots in a class, , is equivalent to the number of walks of length h satisfying the boundary conditions and containing steps ; it appears as

The symmetry of a knot, our first aesthetic constraint, is the number of moves to the right minus the number of moves to the left, i.e.,

where if the ith step is , -1 if the ith step is and 0 otherwise. Since asymmetrical knots disrupt the bilateral symmetry of man, we limit our attention to the most symmetric knots from each class, i.e., those which minimise s.

Whereas the centre number and the symmetry s tell us the move composition of a knot, balance relates to the distribution of these moves; it corresponds to the extent to which the moves are well mixed. A balanced knot is tightly bound and keeps its shape. We use it as our second aesthetic constraint. The balance b may be expressed

where represents the ith step of the walk and the winding direction is equal to 1 if the transition from to is, say, clockwise and -1 otherwise. Of those knots which are optimally symmetric, we desire that knot which minimises b.

The ten canonical knot classes and the corresponding most aesthetic knots are listed in Table 1. The four named knots are the only ones, to our knowledge, to have received widespread attention, either published or through tradition. Unnamed knots are hereby introduced by the authors.

The first four columns describe the knot class , while the remainder relate to the corresponding most aesthetic knot. The center fraction provides a guide to knot shape, the higher fractions corresponding to broader knots; it, along with the size h, should be used in selecting a knot.

Certain readers may observe the use of knots whose sequences are equivalent to those shown in Table 1 apart from transpositions of groups, for instance, the use of in place of the Half-Windsor (T. P. Harte and L. S. G. E. Howard, personal communication); some will argue that this is the Half-Windsor. Such ambiguity follows from the variable width of conventional ties -- the earliest ties were uniformly wide. This makes some transpositions arguably favourable, namely the last group in the knots in Table 1. We do not attempt to distinguish between these knots and their counterparts; this much we leave to the sartorial discretion of the reader.

Thomas M. A. Fink, Yong Mao

Theory of Condensed Matter, Cavendish Laboratory, Cambridge CB3 0HE, UK

e-mail: tmf20@cus.cam.ac.uk, ym101@phy.cam.ac.uk

Requests for materials can be made at: http://www.tcm.phy.cam.ac.uk/ `~`ym101/).

{ All diagrams are drawn in the frame of reference of the mirror image of the actual tie. (top left) The two ways of beginning a knot, and . For knots beginning with , the tie must begin inside out. (bottom) The Four-in-Hand, denoted by the sequence . (top right) A tie knot may be represented by a persistent random walk on a triangular lattice. Shown here is the Four-in-Hand, indicated by the walk .

 h s b Name Sequence 3 1 0.33 1 0 0 4 1 0.25 1 -1 1 Four-in-Hand 5 2 0.40 2 -1 0 Pratt Knot 6 2 0.33 4 0 0 Half-Windsor 7 2 0.29 6 -1 1 7 3 0.43 4 0 1 8 2 0.25 8 0 2 8 3 0.38 12 -1 0 Windsor 9 3 0.33 24 0 0 9 4 0.44 8 -1 2

Aesthetic tie knots, characterised, from left, by half-winding number h, centre number , centre fraction , knots per class , symmetry s, balance b, name and sequence.