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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
1, 0, 1, 1, 3, 3, 12, 19, 64, 155, 510, 1514, 5146, 16966, 58782, 203269, 716607, 2536201, 9062402, 32533568, 117498072, 426212952, 1553048548, 5681011890, 20858998805, 76850220654, 284057538480, 1053134292253, 3915683667721 (list; graph; refs; listen; history; text; internal format)



Number of basic polyhedra with n vertices.

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.

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


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.


Table of n, a(n) for n=6..34.

G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54.

Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.

Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]

A. Caudron, Classification des noeuds et des enlacements, Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982.

Alain Caudron, Classification des noeuds et des enlacements (Thèse et additifs), Univ. Paris-Sud, 1989 [Scanned copy, included with permission]. Contains additional material.

CombOS - Combinatorial Object Server, generate planar graphs

S. V. Jablan, Ordering Knots

S. V. Jablan, L. M. Radović, and R. Sazdanović, Basic polyhedra in knot theory Kragujevac J. Math., 28 (2005), 155-164.

The Knot Atlas, Conway Notation.

Index entries for sequences related to knots


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

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


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

Sequence in context: A073055 A075780 A292515 * A290438 A006804 A052533

Adjacent sequences:  A078663 A078664 A078665 * A078667 A078668 A078669




Slavik V. Jablan and Brendan McKay Feb 06 2003


Name and offset corrected by Andrey Zabolotskiy, Aug 22 2017



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Last modified August 9 20:09 EDT 2020. Contains 336326 sequences. (Running on oeis4.)