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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).
7

%I #58 Oct 24 2024 12:33:07

%S 0,0,1,1,2,3,10,27,101,364,1610,7202,34659,170692,864590,4463287,

%T 23415443,124526110,670224294,3644907768,20011145443,110794212315,

%U 618187581204

%N 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).

%C Irreducible means not made disconnected by removal of a vertex (no nugatory crossings).

%C Indecomposable (or prime) means not made disconnected by cutting two disjoint lines.

%C From _Robert Coquereaux_ and _Andrey Zabolotskiy_, Nov 30 2021: (Start)

%C Equivalently, the number of projections of prime alternating knots with n crossings, or prime knot shadows.

%C 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.

%C A 1999 unpublished result by J. Hoste gives a(15) = 864127, a(16) = 4463287, a(17) = 23415443.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)

%C Using plantri I find a(15) = 864590, agreeing with Bétréma. - _Brendan McKay_, Mar 13 2023

%H Brian Arnold, Michael Au, Christoper Candy, Kaan Erdener, James Fan, Richard Flynn, Robs John Muir, Danny Wu and Jim Hoste, <a href="https://doi.org/10.1142/S0218216594000319">Tabulating alternating knots through 14 crossings</a>, Journal of Knot Theory and Its Ramifications, 3 (1994), 433-437. Gives the sequence up to n = 14.

%H J. Betrema, <a href="https://github.com/j2b2/TaitCurves">Tait Curves</a>

%H Gunnar Brinkmann and Brendan McKay, <a href="https://users.cecs.anu.edu.au/~bdm/plantri/">plantri plane graph generator</a>. 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.

%H Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/1507.03163">Maps, immersions and permutations</a>, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also <a href="https://doi.org/10.1142/S0218216516500474">J. Knot Theory Ramifications</a> (2016) Vol. 25, No. 8, 1650047. Gives the sequence up to n = 10. The immersions for n = 8, 9 are shown in Figs. 15-17.

%H Noboru Ito and Yusuke Takimura, <a href="https://arxiv.org/abs/2108.09698">The tabulation of prime knot projections with their mirror images up to eight double points</a>, Topol. Proc. 53, 177-199 (2019). [The diagrams up to n = 8 are given in Table 4. Reference supplied by K. A. Perko, Jr., Jun 09 2019]

%H Abdullah Khan, Alexei Lisitsa, Viktor Lopatkin and Alexei Vernitski, <a href="https://arxiv.org/abs/2108.02873">Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration</a>, arXiv:2108.02873 [math.GT], 2021.

%H Abdullah Khan, Alexei Lisitsa, and Alexei Vernitski, <a href="https://arxiv.org/abs/2103.02102">Experimental Mathematics Approach to Gauss Diagrams Realizability</a>, arXiv:2103.02102 [math.GT], 2021. Gives the sequence up to n = 13.

%H Abdullah Khan, Alexei Lisitsa, and Alexei Vernitski, <a href="https://doi.org/10.1007/978-3-030-81097-9_16">Gauss-Lintel, an Algorithm Suite for Exploring Chord Diagrams</a>, Intelligent Computer Mathematics, Int'l Conf. Intel. Comp. Math. (CICM 2021), 197-202.

%H T. P. Kirkman, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/enumerat.pdf">The enumeration, description and construction of knots of fewer than ten crossings</a>, Trans. Roy. Soc. Edinburgh 32 (1885), 281-309, <a href="https://doi.org/10.1017/S0080456800026788">doi:10.1017/S0080456800026788</a>.

%H Alexei Lisitsa and Alexei Vernitski, <a href="https://doi.org/10.1016/j.exco.2024.100162">Counting graphs induced by Gauss diagrams and families of mutant alternating knots</a>, Examples Counterex. (2024) Vol. 6, Art. No. 100162.

%H C. N. Little, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/alternat.pdf">Alternate +/- knots of order eleven</a>, Trans. Roy. Soc. Edinburgh 36 (1890), 253-255, <a href="https://doi.org/10.1017/S008045680003773X">doi:10.1017/S008045680003773X</a>.

%H P. G. Tait, <a href="https://doi.org/10.1017/S0080456800090633">On knots</a>, Trans. Roy. Soc. Edin. 28 (1876/77), 145-190.

%o (C) See the J. Betrema C program in the Tait Curves link.

%Y Cf. A008986, A008987, A008988, A008989, A007756, A264760, A264761.

%K nonn,more,changed

%O 1,5

%A _Robert Coquereaux_, Nov 23 2015

%E a(15)-a(21) from _Brendan McKay_, Mar 12 2023

%E Comment on link to plantri modified by _Brendan McKay_, Mar 25 2024

%E a(22) and a(23) from _Brendan McKay_, Mar 30 2024