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A264759
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Number of irreducible indecomposable spherical curves with n crossings (only ordinary double points), the circle is not oriented, the sphere is not oriented (UU case).
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6
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0, 0, 1, 1, 2, 3, 10, 27, 101, 364, 1610, 7202, 34659, 170692, 864590, 4463287, 23415443, 124526110, 670224294, 3644907768, 20011145443, 110794212315, 618187581204
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OFFSET
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1,5
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COMMENTS
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Irreducible means not made disconnected by removal of a vertex (no nugatory crossings).
Indecomposable (or prime) means not made disconnected by cutting two disjoint lines.
Equivalently, the number of projections of prime alternating knots with n crossings, or prime knot shadows.
This sequence up to n = 10 was known to Kirkman (1885) and confirmed by Little (1890). The terms up to n = 14 are given by Hoste et al. (1994) and independently found by J. Bétréma using his program.
A 1999 unpublished result by J. Hoste gives a(15) = 864127, a(16) = 4463287, a(17) = 23415443. J. Bétréma's program gives the same a(16) but different a(15) = 864590. (End)
Using plantri I find a(15) = 864590, agreeing with Bétréma. - Brendan McKay, Mar 13 2023
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LINKS
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Brian Arnold, Michael Au, Christoper Candy, Kaan Erdener, James Fan, Richard Flynn, Robs John Muir, Danny Wu and Jim Hoste, Tabulating alternating knots through 14 crossings, Journal of Knot Theory and Its Ramifications, 3 (1994), 433-437. Gives the sequence up to n = 14.
Gunnar Brinkmann and Brendan McKay, plantri plane graph generator. To obtain this sequence use options -Guqc2m2d (which makes plane quartic graphs) and count those for which the straight-ahead Eulerian walk has a single component.
P. G. Tait, On knots, Trans. Roy. Soc. Edin. 28 (1876/77), 145-190.
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PROG
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(C) See the J. Betrema C program in the Tait Curves link.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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Comment on link to plantri modified by Brendan McKay, Mar 25 2024
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STATUS
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approved
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