

A122495


Integers corresponding to rational knots in Conway's enumeration.


2



1, 3, 22, 5, 32, 42, 312, 2112, 7, 52, 43, 322, 313, 2212, 21112, 62, 512, 44, 413, 4112, 332, 3212, 3113, 31112, 2312, 2222, 22112, 9, 72, 63, 54, 522, 513, 423, 4212, 4122, 41112, 342, 333, 3222, 3213, 31212, 31122, 311112, 2412, 2322, 23112, 22122, 21312
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

"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 claims to have verified the TaitLittle tables 'in an afternoon'.
"He then went on to list the 100crossings knots and 10crossing 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, 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.
"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 2component 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).
"This shows that the only amphicheiral knots in the list are the figure8 knot (sequence 22) and the knot 6_3 (sequence 2112); all of the links are cheiral...." [Cromwell]
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 1digit, it's not a problem, but a(97) requires "digit" 11, so at that point the sequence becomes not fully welldefined. An irregular array of these numbers would be welldefined.  Andrey Zabolotskiy, May 22 2017


REFERENCES

Peter R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 209211.


LINKS

Table of n, a(n) for n=1..49.
Alain Caudron, Classification des noeuds et des enlancements (see p. 168).
J. H. Conway, An enumeration of knots and links and some of their algebraic properties, 1970. Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) pp. 329358 Pergamon, Oxford.
Index entries for sequences related to knots


EXAMPLE

a(1) = 1 because 1 corresponds to the trivial knot.
a(2) = 3 because 3 corresponds to the trefoil.
a(3) = 22 because 22 corresponds to the figure8 knot.


MATHEMATICA

whereTangle[{n_}] := If[EvenQ[n], 1, 2];
whereTangle[{rest__, n_}] := Switch[whereTangle[{rest}], 1, 3, 2, Switch[whereTangle[{n}], 1, 2, 2, 1, 3, 3], 3, whereTangle[{n}]];
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 2component links*) &], {1}]
(* Andrey Zabolotskiy, May 22 2017 *)


CROSSREFS

Cf. A002863, A002864, A173637, A018240, A078666.
Sequence in context: A277628 A016449 A161377 * A100977 A037101 A226028
Adjacent sequences: A122492 A122493 A122494 * A122496 A122497 A122498


KEYWORD

nonn


AUTHOR

Jonathan Vos Post, Sep 16 2006


EXTENSIONS

Sequence edited and more terms added by Andrey Zabolotskiy, May 22 2017


STATUS

approved



