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COMMENT
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"Conway's motivation for studying tangles was to extend the [knot and link] catalogues.... here we shall concentrate on finding the first few rational links.
"The problem is reduced to listing sequences of integers, and noting which sequences lead to isotopic links.
"The technique is so powerful that Conway vlaims to have verified the Tait-Little tables 'in an afternoon'.
"He then went on to list the 100-crossings knots and 10-crossing links.... A rational link (or its mirror image) has a regular continued fraction expansion in which all the integers are positive....
"We can discard all sequences that end in a 1, and that makes the regular sequence unique.... we do not need to keep both a sequence and its reverse.
"Applying these simple rules to the partitions of the first four integers, we see that we keep only the sequences shown in bold: 1, 2, 11, 3, 21, 12, 111, 4, 31, 22, 13, 211, 121, 112, 1111." [typographically, the bold subsequence is 1, 2, 4, 22] "These sequences correspond to the trvial knot, the Hopf link, the trefoil, the (2,4) torus link, and the figure 8 knot.
"Continuing in this fashion, we find that for knots and links with up to seven crossings, the sequence for rational knots are: 3, 22, 5, 32, 42, 312, 2112, 7, 52, 43, 322, 313, 2212, 21112 and the sequences for rational 2-component links are 2, 4, 212, 6, 33, 222, 412, 232, 3112.... we see that a sequence represents an amphicheiral knot or link only if the sequence is is palindromic (equal to its reverse) and of even length (n even).
"This shows that the only amphicheiral knots in the list are the figure-8 knot (sequence 22) and the knot 6_3 (sequence 2112); all of the links are cheiral...." [Cromwell]
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