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 A078666 Number of isomorphism classes of simple quadrangulations of the sphere having n+2 vertices and n faces, minimal degree 3, with orientation-reversing isomorphisms permitted. 13

%I

%S 1,0,1,1,3,3,12,19,64,155,510,1514,5146,16966,58782,203269,716607,

%T 2536201,9062402,32533568,117498072,426212952,1553048548,5681011890,

%U 20858998805,76850220654,284057538480,1053134292253,3915683667721

%N Number of isomorphism classes of simple quadrangulations of the sphere having n+2 vertices and n faces, minimal degree 3, with orientation-reversing isomorphisms permitted.

%C Number of basic polyhedra with n vertices.

%C Initial terms of sequence coincide with A007022. Starting from n=12, to it is added the number of simple 4-regular 4-edge-connected but not 3-connected plane graphs on n nodes (A078672). As a result we obtain the number of basic polyhedra.

%C a(n) counts 4-valent 4-edge-connected planar maps (or plane graphs on a sphere) up to reflection with no regions bounded by just 2 edges. Conway called such maps "basic polyhedra" and used them in his knot notation. 2-edge-connected maps (which start occurring from n=12) are not taken into account here because they generate only composite knots and links. - _Andrey Zabolotskiy_, Sep 18 2017

%D J. H. Conway, An enumeration of knots and links and some of their related properties. Computational Problems in Abstract Algebra, Proc. Conf. Oxford 1967 (Ed. J. Leech), 329-358. New York: Pergamon Press, 1970.

%H G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, <a href="http://dx.doi.org/10.1016/j.disc.2005.10.005">Generation of simple quadrangulations of the sphere</a>, Discr. Math., 305 (2005), 33-54.

%H Gunnar Brinkmann and Brendan McKay, <a href="http://users.cecs.anu.edu.au/~bdm/plantri/">plantri and fullgen</a> programs for generation of certain types of planar graph.

%H Gunnar Brinkmann and Brendan McKay, <a href="/A000103/a000103_1.pdf">plantri and fullgen</a> programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]

%H A. Caudron, <a href="http://sites.mathdoc.fr/PMO/PDF/C_CAUDRON_82_04.pdf">Classification des noeuds et des enlacements</a>, Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982.

%H Alain Caudron, <a href="/A002863/a002863_3.pdf">Classification des noeuds et des enlacements (Thèse et additifs)</a>, Univ. Paris-Sud, 1989 [Scanned copy, included with permission]. Contains additional material.

%H CombOS - Combinatorial Object Server, <a href="http://combos.org/plantri">generate planar graphs</a>

%H S. V. Jablan, <a href="http://members.tripod.com/vismath/sl/index.html">Ordering Knots</a>

%H S. V. Jablan, L. M. Radović, and R. Sazdanović, <a href="http://eudml.org/doc/253048">Basic polyhedra in knot theory</a> Kragujevac J. Math., 28 (2005), 155-164.

%H The Knot Atlas, <a href="http://katlas.org/wiki/Conway_Notation">Conway Notation</a>.

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

%e G.f. = x^6 + x^8 + x^9 + 3*x^10 + 3*x^11 + 12*x^12 + 19*x^13 + 64*x^14 + ...

%e a(6)=1, a(7)=0, a(8)=1, a(9)=1, a(10)=3, etc.

%Y Cf. A007022, A078672, A113201, A072552, A292515 (planar graphs with same restrictions).

%K nonn

%O 6,5

%A Slavik V. Jablan and _Brendan McKay_ Feb 06 2003

%E Name and offset corrected by _Andrey Zabolotskiy_, Aug 22 2017

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Last modified September 28 13:24 EDT 2020. Contains 337393 sequences. (Running on oeis4.)