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A122495 Integers corresponding to rational knots in Conway's enumeration. 2


%S 1,3,22,5,32,42,312,2112,7,52,43,322,313,2212,21112,62,512,44,413,

%T 4112,332,3212,3113,31112,2312,2222,22112,9,72,63,54,522,513,423,4212,

%U 4122,41112,342,333,3222,3213,31212,31122,311112,2412,2322,23112,22122,21312

%N Integers corresponding to rational knots in Conway's enumeration.

%C "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.

%C "The problem is reduced to listing sequences of integers and noting which sequences lead to isotopic links.

%C "The technique is so powerful that Conway claims to have verified the Tait-Little tables 'in an afternoon'.

%C "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....

%C "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.

%C "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, 3, 4, 22] "These sequences correspond to the trivial knot, the Hopf link, the trefoil, the (2,4) torus link and the figure 8 knot.

%C "Continuing in this fashion, we find that for knots and links with up to seven crossings, the sequences 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 palindromic (equal to its reverse) and of even length (n even).

%C "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]

%C The ordering among the terms with the same sum of digits (i.e., number of crossings) is the inverse lexicographical. Each term is actually an ordered set of positive integers, concatenated; as long as all integers are 1-digit, it's not a problem, but a(97) requires "digit" 11, so at that point the sequence becomes not fully well-defined. An irregular array of these numbers would be well-defined. - _Andrey Zabolotskiy_, May 22 2017

%D Peter R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 209-211.

%H Alain Caudron, <a href="http://sites.mathdoc.fr/PMO/PDF/C_CAUDRON_82_04.pdf">Classification des noeuds et des enlancements</a> (see p. 168).

%H J. H. Conway, <a href="http://www.maths.ed.ac.uk/~aar/papers/conway.pdf">An enumeration of knots and links and some of their algebraic properties</a>, 1970. Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) pp. 329-358 Pergamon, Oxford.

%H <a href="/index/K#knots">Index entries for sequences related to knots</a>

%e a(1) = 1 because 1 corresponds to the trivial knot.

%e a(2) = 3 because 3 corresponds to the trefoil.

%e a(3) = 22 because 22 corresponds to the figure-8 knot.

%t whereTangle[{n_}] := If[EvenQ[n], 1, 2];

%t whereTangle[{rest__, n_}] := Switch[whereTangle[{rest}], 1, 3, 2, Switch[whereTangle[{n}], 1, 2, 2, 1, 3, 3], 3, whereTangle[{n}]];

%t FromDigits /@ Prepend[Select[Flatten[Table[Reverse@SortBy[Flatten[Permutations /@ IntegerPartitions[n], 1], PadRight[#, n] &], {n, 10}], 1], OrderedQ[{Reverse[#], #}] && Last[#] != 1 && whereTangle[#] != 1 (*change to "==1" for rational 2-component links*) &], {1}]

%t (* _Andrey Zabolotskiy_, May 22 2017 *)

%Y Cf. A002863, A002864, A173637, A018240, A078666.

%K nonn

%O 1,2

%A _Jonathan Vos Post_, Sep 16 2006

%E Sequence edited and more terms added by _Andrey Zabolotskiy_, May 22 2017

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Last modified February 21 04:18 EST 2019. Contains 320371 sequences. (Running on oeis4.)