%I #28 May 14 2024 00:47:55
%S 0,0,0,0,0,1,0,1,1,3,3,12,19,63,153,499,1473,4974,16296,56102,192899,
%T 674678,2381395,8468424
%N Number of 4-regular 4-edge-connected planar simple graphs on n vertices.
%C The difference between this sequence and A078666 arises because the latter lists not abstract planar graphs but plane graphs (on the sphere, with the same restrictions). Among A078666(14)=64 plane graphs there is 1 pair of isomorphic graphs, namely graphs No. 63 and 64 (hereafter the enumeration of plane graphs from the LinKnot Mathematica package is used, see The Knot Atlas link), hence a(14)=64-1=63. Among 155 plane graphs on 15 vertices, the isomorphic pairs are (143, 149) and (153, 155), hence a(15)=155-2=153. The 11 isomorphic pairs of plane graphs on 16 vertices are: (456, 492), (459, 493), (464, 496), (465, 501), (466, 468), (470, 487), (473, 503), (477, 488), (478, 479), (486, 497), (498, 504).
%C Tuzun and Sikora say that such planar graphs constitute the set of 4-edge-connected basic Conway polyhedra, and indeed it suffices to consider any one embedding of each of these graphs into sphere or plane to list all prime knots. However, usually the set of Conway polyhedra is identified with the set of plane graphs instead (see A078666 and references therein), which is necessary to list or encode all prime knot diagrams (on the sphere).
%H The Knot Atlas, <a href="http://katlas.org/wiki/Conway_Notation">Conway Notation</a>.
%H Robert E. Tuzun and Adam S. Sikora, <a href="https://doi.org/10.1142/S0218216518400096">Verification Of The Jones Unknot Conjecture Up To 22 Crossings</a>, Journal of Knot Theory and Its Ramifications (2018) 1840009, <a href="https://arxiv.org/abs/1606.06671">arXiv:1606.06671</a> [math.GT], 2016-2020 (see table 2).
%H Robert E. Tuzun and Adam S. Sikora, <a href="https://arxiv.org/abs/2003.06724">Verification Of The Jones Unknot Conjecture Up To 24 Crossings</a>, arXiv:2003.06724 [math.GT], 2020 (see table 1).
%e From _Allan Bickle_, May 13 2024: (Start)
%e For n=6, the unique graph is the octahedron.
%e For n=8, the unique graph is the square of an 8-cycle.
%e For n=9, the unique graph is the dual of the Herschel graph. (End)
%Y Cf. A007022, A072552, A078666, A111361 (4-regular planar graphs with restrictions).
%K nonn,more
%O 1,10
%A _Andrey Zabolotskiy_, Sep 18 2017
%E a(23)-a(24) added from Tuzun & Sikora (2020) by _Andrey Zabolotskiy_, Apr 27 2020