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A000241
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Crossing number of complete graph with n nodes. Dubious for n >= 13.
(Formerly M2772 N1115)
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9
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0, 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, 150, 225, 315, 441, 588
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| Verified for n=11, 12 by Shengjun Pan and R. Bruce Richter, in "The Crossing Number of K_11 is 100", submitted. Still dubious for n >= 13.
Also the sum of the dimensions of the irreducible representations of su(3) that first occur in the [n-5]th tensor power of the tautological representation. - james dolan (jdolan(AT)math.ucr.edu), Jun 02 2003
It appears that a(n)=C(floor(n/2),2)*C(floor((n-1)/2),2). [From Paul Barry (pbarry(AT)wit.ie), Oct 02 2008]
Contribution from Paul Barry (pbarry(AT)wit.ie), Oct 02 2008: (Start)
We conjecture that this sequence is given by one half of the third coefficient
of the denominator polynomial of the n-th convergent to the g.f. of n!,
in which case the next numbers are 784,1008,1296,1620, 2025, 2475,...
Essentially sum{k=0..n, (-1)^(n-k) floor(k/2)ceiling(k/2)floor((k-1)/2)ceiling((k-1)/2)/2}. (End)
One of the most basic questions in knot theory remains unresolved: is crossing number additive under connected sum? In other words, does the equality c(K1#K2) = c(K1) + c(K2) always hold, where c(K) denotes the crossing number of a knot K and K1#K2 is the connected sum of two (oriented) knots K1 and K2? Theorem 1.1. Let K1, . . .,Kn be oriented knots in the 3-sphere. Then (c(K1) + . . . + c(Kn)) / 152 <= c(K1# . . . #Kn) <= c(K1) + . . . + c(Kn). [From Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 26 2009]
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REFERENCES
| P. Erdos and R. K. Guy, Crossing number problems, Amer. Math. Monthly, 80 (1973), 52-58.
R. K. Guy, The crossing number of the complete graph, Bull. Malayan Math. Soc., Vol. 7, pp. 68-72, 1960.
D. McQuillan and R. B. Richter, A parity theorem for drawings of complete ... graphs, Amer. Math. Monthly, 117 (2010), 267-273.
A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277-280.
T. L. Saaty, The number of intersections in complete graphs, Engrg. Cybernetics 9 (1971), no. 6, 1102-1104 (1972).; translated from Izv. Akad. Nauk SSSR Tehn. Kibernet. 1971, no. 6, 151-154 (Russian). Math. Rev. 58 #21749.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.
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LINKS
| Drago Bokal, Gasper Fijavz and David R. Wood, The Minor Crossing Number of Graphs with an Excluded Minor, math.CO/0609707.
J. Dolan et al., su(3) and Zarankiewicz's conjecture
Marc Lackenby, The crossing number of composite knots, Aug 25, 2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 26 2009]
T. L. Saaty, The Minimum Number Of Intersections In Complete Graphs
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
E. Weisstein, Zarankiewicz's Conjecture.html
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CROSSREFS
| It is not known if A000241 and A028723 agree. Cf. A007333, A014540, A030179.
Cf. A121021, A191928.
Sequence in context: A093446 A132920 A127645 * A028723 A057578 A015635
Adjacent sequences: A000238 A000239 A000240 * A000242 A000243 A000244
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Bokal et al. link from Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 08 2006
New arXiv citation. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 26 2009]
ArXiv URL replaced by non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 23 2009
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