## Conclusion

Table 1 lists the ten desired knot classes and the corresponding most aesthetic knots. The four named knots are the only ones, to our knowledge, to have received widespread attention, either published or through tradition; indeed, we have never observed any of the other six (although it has been brought to our attention that the first entry, , finds widespread use throughout the communist youth organisation of China).

The first four columns describe the knot class , while the middle three relate to the corresponding most aesthetic knot. For qualities extrinsic to knot size h, namely and b, we provide the analogous intrinsic qualities as well. 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 will have observed the use of knots whose sequences are equivalent to those shown below apart from transpositions of groups, for instance, the use of in place of the Half-Windsor; indeed, 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), which makes some transpositions arguably favourable. We make no attempt to address this point; at last we call upon the sartorial discretion of the reader.

 h s b b/h Name Sequence Knotted 3 1 0.33 1 0 0 0 y 4 1 0.25 1 -1 1 0.25 Four-in-Hand n 5 2 0.40 2 -1 0 0 Pratt Knot n 6 2 0.33 4 0 0 0 Half-Windsor y 7 2 0.29 6 -1 1 0.14 n 7 3 0.43 4 0 1 0.14 y 8 2 0.25 8 0 2 0.25 y 8 3 0.38 12 -1 0 0 Windsor n 9 3 0.33 24 0 0 0 y 9 4 0.44 8 -1 2 0.22 n

Table i: Aesthetic tie knots, characterised, from left, by half-winding number h, centre number , centre fraction , possible knots per class , symmetry s, balance b, balance fraction , name, sequence and knotted status. Unnamed knots are hereby introduced by the authors.

We acknowledge Dr. R. C. Ball, Prof. M. E. Cates, T. P. Harte and Dr. L. S. G. E. Howard for stimulating discussion on the subject. T. M. F. is supported by a National Science Foundation Graduate Research Fellowship. Y. M. is supported by a Research Studentship from Trinity College, Cambridge.

Correspondence and requests for materials should be addressed to (e-mail: ym101@cus.cam.ac.uk, internet: http://www.tcm.phy.cam.ac.uk/`~`ym101/).