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Conclusion

Table 1 lists the ten desired knot classes tex2html_wrap_inline898 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, tex2html_wrap_inline796 , finds widespread use throughout the communist youth organisation of China).

The first four columns describe the knot class tex2html_wrap_inline898 , while the middle three relate to the corresponding most aesthetic knot. For qualities extrinsic to knot size h, namely tex2html_wrap_inline800 and b, we provide the analogous intrinsic qualities as well. The center fraction tex2html_wrap_inline802 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 tex2html_wrap_inline1292 groups, for instance, the use of tex2html_wrap_inline1294 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 tex2html_wrap_inline1296 tex2html_wrap_inline1298 tex2html_wrap_inline804 s b b/h Name Sequence Knotted
3 1 0.33 1 0 0 0 tex2html_wrap_inline796 y
4 1 0.25 1 -1 1 0.25 Four-in-Hand tex2html_wrap_inline780 n
5 2 0.40 2 -1 0 0 Pratt Knot tex2html_wrap_inline1312 n
6 2 0.33 4 0 0 0 Half-Windsor tex2html_wrap_inline1316 y
7 2 0.29 6 -1 1 0.14 tex2html_wrap_inline1320 n
7 3 0.43 4 0 1 0.14 tex2html_wrap_inline1324 y
8 2 0.25 8 0 2 0.25 tex2html_wrap_inline1328 y
8 3 0.38 12 -1 0 0 Windsor tex2html_wrap_inline1332 n
9 3 0.33 24 0 0 0 tex2html_wrap_inline1336 y
9 4 0.44 8 -1 2 0.22 tex2html_wrap_inline1340 n

 


Table i: Aesthetic tie knots, characterised, from left, by half-winding number h, centre number tex2html_wrap_inline800 , centre fraction tex2html_wrap_inline802 , possible knots per class tex2html_wrap_inline804 , symmetry s, balance b, balance fraction tex2html_wrap_inline810 , 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/).


next up previous
Next: About this document Up: Tie Knots and Random Previous: Knotted or Unknotted

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