Original entry on oeis.org
1, 0, 1, 1, 2, 3, 10, 27, 101, 364, 1562
Offset: 1
Slavik Jablan and Radmila Sazdanovic, Jan 08 2004
- Ito, Noboru; and Takimura, Yusuke, The tabulation of prime knot projections with their mirror images up to eight double points. Topol. Proc. 53, 177-199 (2019). [Reference supplied by K. A. Perko, Jr., Jun 09 2019]
- Kirkman T.P.: The enumeration, description and construction of knots of fewer than ten crossings. Trans. Roy. Soc. Edinburgh 32 (1885), 281-309.
- Tait P.G.: On knots. Trans. Roy. Soc. Edin. 28 (1876/77), 145-190.
A008989
Number of immersions of an unoriented circle into the unoriented sphere with n double points.
Original entry on oeis.org
1, 1, 2, 6, 19, 76, 376, 2194, 14614, 106421, 823832, 6657811, 55557329, 475475046, 4155030702, 36959470662, 333860366236
Offset: 0
G.f. = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 76*x^5 + 376*x^6 + 2194*x^7 + ...
- V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994, p. 18.
- J. Cantarella, H. Chapman, and M. Mastin, Knot Probabilities in Random Diagrams, arXiv preprint arXiv:1512.05749 [math.GT], 2015. Also Journal of Physics A: Mathematical and Theoretical, Vol. 49, No. 40 (2016), DOI: 10.1088/1751-8113/49/40/405001
- R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), 10.1142/S0218216516500474.
- Guy Valette, A Classification of Spherical Curves Based on Gauss Diagrams, Arnold Math J. (2016) 2:383-405.
A264760
Number of irreducible indecomposable spherical curves with n crossings (only ordinary double points), the circle is not oriented, the sphere is oriented (UO case).
Original entry on oeis.org
0, 0, 1, 1, 2, 4, 12, 41, 161, 658, 2993, 13974, 67945, 338644, 1720544, 8908579, 46775073, 248932094, 1340079951, 7289000415, 40019815872, 221582832331, 123635832467
Offset: 1
- J. Betrema, Tait Curves
- R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: http://dx.doi.org/10.1142/S0218216516500474
- Gunnar Brinkmann and Brendan McKay, plantri plane graph generator. To obtain this sequence use options -Guoqc2m2d (which makes plane quartic graphs) and count those for which the straight-ahead Eulerian walk has a single component.
A264761
Number of irreducible indecomposable spherical curves with n crossings (only ordinary double points), the circle is oriented, the sphere is oriented (OO case).
Original entry on oeis.org
0, 0, 1, 1, 2, 6, 17, 73, 290, 1274, 5844, 27750, 135192, 676263, 3437509, 17811771, 93531354, 497835030, 2680058068, 14577839412, 80039070868, 443164758244, 2472713506356
Offset: 1
A007756
Number of irreducible indecomposable spherical curves with n crossings (only ordinary double points), the circle is oriented, the sphere is not oriented (OU case).
Original entry on oeis.org
0, 0, 1, 1, 2, 3, 11, 38, 156, 638, 2973, 13882, 67868, 338147, 1720303, 8905996, 46774728, 248918004, 1340083514, 7288922610, 40019870539, 221582395052, 1236358849827
Offset: 1
- J. Betrema, Tait Curves
- R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: 10.1142/S0218216516500474
- C. Ernst, C. Hart, T. Menezes and D. Price, A complete list of minimal diagrams of an oriented alternating knot, J. Knot Theory Ramifications 30, 2150063 (2021). See section 3.1.
A268571
Number of immersions of unoriented circle into unoriented sphere with n double points and no simple loop.
Original entry on oeis.org
0, 0, 1, 1, 2, 5, 16, 52, 205, 863
Offset: 1
- Robert Coquereaux and Jean-Bernard Zuber, Maps, immersions and permutations, Journal of Knot Theory and Its Ramifications, Vol. 25, No. 8 (2016), 1650047; arXiv preprint, arXiv:1507.03163 [math.CO], 2015-2016. See Table 6.
- Guy Valette, A Classification of Spherical Curves Based on Gauss Diagrams, Arnold Math. J. (2016) 2:383-405. See Table 7.
A343358
Number of connected graphs with n vertices which are realizable (in the sense of realizability of Gauss diagrams).
Original entry on oeis.org
1, 1, 2, 3, 7, 18, 41, 123, 361, 1257, 4573
Offset: 3
- L. Bishler et al. "Distinguishing mutant knots." Journal of Geometry and Physics 159 (2021): 103928.
- L. Bishler, et al., Distinguishing mutant knots, arXiv:2007.12532 [hep-th], 2021.
- Abdullah Khan, Alexei Lisitsa, Viktor Lopatkin, and Alexei Vernitski, Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration, arXiv:2108.02873 [math.GT], 2021.
- Alexei Lisitsa, Abdullah Khan, and Alexei Vernitski, An experimental approach to Gauss diagram realizability, 28th British Comb. Conf., Durham Univ. (UK, 2021), p. 107.
- Alexei Lisitsa and Alexei Vernitski, Counting graphs induced by Gauss diagrams and families of mutant alternating knots, Examples Counterex. (2024) Vol. 6, Art. No. 100162.
- A. Stoimenow, Knot data tables.
Cf.
A002864, which starts with 1, 1, 2, 3, 7, 18, 41, 123, 367. This is because an alternating prime knot with 10 or fewer crossings is uniquely defined by the graph of the corresponding closed planar curve. Only starting from n=11 some alternating knots which share the same graph but are distinct knots (called "mutant knots") start appearing.
Cf.
A264759, which starts with 1, 1, 2, 3, 10; there is a mismatch starting from size 7. Indeed, starting from n=7 there are some planar curves which share the same graph but have distinct Gauss diagrams.
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