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/).