A264761 -id:A264761 - cp's OEIS frontend

A264759 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).

0, 0, 1, 1, 2, 3, 10, 27, 101, 364, 1610, 7202, 34659, 170692, 864590, 4463287, 23415443, 124526110, 670224294, 3644907768, 20011145443, 110794212315, 618187581204

Offset: 1

Robert Coquereaux, Nov 23 2015

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.
From Robert Coquereaux and Andrey Zabolotskiy, Nov 30 2021: (Start)
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A 1999 unpublished result by J. Hoste gives a(15) = 864127, a(16) = 4463287, a(17) = 23415443.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

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.
J. Betrema, Tait Curves
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.
Robert Coquereaux and Jean-Bernard Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications (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.
Noboru Ito and Yusuke Takimura, The tabulation of prime knot projections with their mirror images up to eight double points, 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]
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.
Abdullah Khan, Alexei Lisitsa, and Alexei Vernitski, Experimental Mathematics Approach to Gauss Diagrams Realizability, arXiv:2103.02102 [math.GT], 2021. Gives the sequence up to n = 13.
Abdullah Khan, Alexei Lisitsa, and Alexei Vernitski, Gauss-Lintel, an Algorithm Suite for Exploring Chord Diagrams, Intelligent Computer Mathematics, Int'l Conf. Intel. Comp. Math. (CICM 2021), 197-202.
T. P. Kirkman, The enumeration, description and construction of knots of fewer than ten crossings, Trans. Roy. Soc. Edinburgh 32 (1885), 281-309, doi:10.1017/S0080456800026788.
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.
C. N. Little, Alternate +/- knots of order eleven, Trans. Roy. Soc. Edinburgh 36 (1890), 253-255, doi:10.1017/S008045680003773X.
P. G. Tait, On knots, Trans. Roy. Soc. Edin. 28 (1876/77), 145-190.

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

See the J. Betrema C program in the Tait Curves link.

a(15)-a(21) from Brendan McKay, Mar 12 2023
Comment on link to plantri modified by Brendan McKay, Mar 25 2024
a(22) and a(23) from Brendan McKay, Mar 30 2024

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

Robert Coquereaux, Nov 23 2015

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.

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.

Cf. A008986, A008987, A008988, A008989, A007756, A264759, A264761.

See the J. Betrema C program in the Tait Curves link.

a(14)-a(21) from Brendan McKay, Mar 12 2023
plantri link added by Brendan McKay, Mar 25 2024
a(22) and a(23) from Brendan McKay, Mar 30 2024

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

Jean Betrema

Old name was "Prime Gaussian (i.e. only ordinary double points) curves with n crossings."
Irreducible means not made disconnected by removal of a vertex (no nugatory crossings).
Indecomposable (or prime) means not made disconnected by cutting two distinct lines.

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.

Cf. A008986, A008987, A008988, A008989, A264759, A264760, A264761.

See the J. Betrema C program in the Tait Curves link.

Edited by Robert Coquereaux, Nov 23 2015
a(15)-a(16) from Sean A. Irvine, Jan 22 2018
a(17)-a(23) from Brendan McKay, Mar 30 2024

A268568 Number of immersions of oriented circle into oriented sphere with n double points and no simple loop.

Original entry on oeis.org

0, 0, 1, 1, 2, 9, 29, 133, 594, 2864

Offset: 1

N. J. A. Sloane, Mar 03 2016

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.

Cf. A008986, A264761, A268569, A268570, A268571, A268572.

Name clarified by Andrey Zabolotskiy, Jun 09 2024

A268573 Number of irreducible indecomposable bicolored immersions of unoriented circle into oriented sphere with n double points.

Original entry on oeis.org

0, 0, 2, 1, 4, 6, 24, 73, 322, 1274

Offset: 1

N. J. A. Sloane, Mar 03 2016

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 7.

Cf. A264760, A264761, A268561, A268572.

a(2*n) = A264761(2*n), a(2*n+1) = 2 * A264760(2*n+1). - Andrey Zabolotskiy, Jun 05 2024

Name clarified by Andrey Zabolotskiy, Jun 08 2024

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A264759 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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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).

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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).

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A268568 Number of immersions of oriented circle into oriented sphere with n double points and no simple loop.

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A268573 Number of irreducible indecomposable bicolored immersions of unoriented circle into oriented sphere with n double points.

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