A008987 -id:A008987 - cp's OEIS frontend

A008986 Number of immersions of oriented circle into oriented sphere with n double points.

1, 1, 3, 9, 37, 182, 1143, 7553, 54559, 412306, 3251240, 26478264, 221714164, 1900103364, 16613990484, 147816411921, 1335367515821

Offset: 0

N. J. A. Sloane

a(10) = 3251240 by sampling method (to be confirmed, cf. ref. arXiv:1507.03163). - Robert Coquereaux, Nov 23 2015

a(10) = 3251240 confirmed. - Brendan McKay, Mar 29 2024

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994, p. 18.

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
Guy Valette, A Classification of Spherical Curves Based on Gauss Diagrams, Arnold Math J. (2016) 2:383-405.

Cf. A008987, A008988, A008989. First line of triangle A260285.

Magma
```
/* For Magma program see A260285. */
```

a(6)-a(7) from Guy Valette, Feb 09 2004
a(8)-a(9) from Robert Coquereaux and Jean-Bernard Zuber, Jul 21 2015
a(10) from same source added by N. J. A. Sloane, Mar 03 2016
a(11)-a(16) from Brendan McKay, Mar 29 2024

A008988 Number of immersions of oriented circle into unoriented sphere with n double points.

Original entry on oeis.org

1, 1, 2, 6, 21, 97, 579, 3812, 27328, 206410, 1625916, 13241177, 110859326, 950069179, 8307012899, 73908363060, 667683905600

Offset: 0

N. J. A. Sloane

a(10) = 1625916 by sampling method (to be confirmed, cf. ref. arXiv:1507.03163). - Robert Coquereaux, Nov 23 2015

a(10) = 1625916 confirmed. - Brendan McKay, Mar 29 2024

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994, p. 18.
Guy Valette, A Classification of Spherical Curves Based on Gauss Diagrams, Arnold Math J. (2016) 2:383-405, DOI 10.1007/s40598-016-0049-3.

R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also DOI, J. Knot Theory Ramifications 25, 1650047 (2016).

Cf. A008986, A008987, A008989. First line of triangle A260885.

Magma
```
// see A260885
```

a(6)-a(7) from Guy Valette, Feb 09 2004
a(8)-a(9) from Robert Coquereaux and Jean-Bernard Zuber, Jul 21 2015
a(10) from same source added by N. J. A. Sloane, Mar 03 2016
a(11)-a(16) from Brendan McKay, Mar 29 2024

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

N. J. A. Sloane

			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.

Cf. A008986, A008987, A008988, A264759, A277739. First line of triangle A260914.

Magma
```
// see A260914
```

a(6)-a(7) from Guy Valette, Feb 09 2004
a(8)-a(9) from Robert Coquereaux and Jean-Bernard Zuber, Jul 21 2015
a(10) from same source added by N. J. A. Sloane, Mar 03 2016
a(11)-a(14) from Brendan McKay, Mar 11 2023
a(15)-a(16) from Brendan McKay, Mar 29 2024

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).

Original entry on oeis.org

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

A260848 Triangle read by rows: T(n,g) = number of general immersions of a circle with n crossings in a surface of arbitrary genus g (the circle is not oriented, the surface is oriented).

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 21, 64, 36, 0, 99, 559, 772, 108, 0, 588, 5656, 14544, 7222, 0, 0, 3829, 56528, 246092, 277114, 34680, 0, 0, 27404, 581511, 3900698, 8180123, 3534038, 0, 0, 0, 206543, 6020787, 58838383, 203964446, 198551464, 22521600, 0, 0, 0

Offset: 1

Robert Coquereaux, Aug 01 2015

When transposed, displayed as an upper right triangle, the first line g = 0 of the table gives the number of immersions of a circle with n double points in a sphere (spherical curves) starting with n=1, the second line g = 1 gives immersions in a torus, etc.
Row g=0 is A008987 starting with n = 1.
For g > 0 the immersions are understood up to stable geotopy equivalence (the counted curves cannot be immersed in a surface of smaller genus). - Robert Coquereaux, Nov 23 2015

			The transposed triangle starts:
1    2   6   21    99     588      3829      27404            206543
     1   6   64   559    5656     56528     581511           6020787
         1   36   772   14544    246092    3900698          58838383
              0   108    7222    277114    8180123         203964446
                    0      0      34680    3534038         198551464
                           0          0         0           22521600
                                      0         0                  0
                                                0                  0

R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv:1507.03163 [math.CO], 2015, Table 9.

The sum over all genera g for a fixed number n of crossings is given by sequence A260847.

Cf. A008987, A260285, A260885, A260914.

/* Example n := 6 */
n:=6;
n; // n: number of crossings
G:=Sym(2*n);
doubleG := Sym(4*n);
genH:={};
for j in [1..(n-1)] do v := G!(1,2*j+1)(2, 2*j+2); Include(~genH,v) ; end for;
H := PermutationGroup< 2*n |genH>; //  The H=S(n) subgroup of S(2n)
cardH:=#H;
cardH;
rho:=Identity(G); for j in [0..(n-1)] do v := G!(2*j+1, 2*j+2) ; rho := rho*v ; end for;
cycrho := PermutationGroup< 2*n |{rho}>; // The cyclic subgroup Z2 generated by rho (mirroring)
Hcycrho:=sub;  // The subgroup generated by H and cycrho
cardZp:= Factorial(2*n-1);
beta:=G!Append([2..2*n],1); // A typical circular permutation
Cbeta:=Centralizer(G,beta);
bool, rever := IsConjugate(G,beta,beta^(-1));
cycbeta := PermutationGroup< 2*n |{rever}>;
Cbetarev := sub;
psifct := function(per);
perinv:=per^(-1);
res:= [IsOdd(j) select (j+1)^per  else j-1 + 2*n : j in [1..2*n] ];
resbis := [IsOdd((j-2*n)^perinv) select  (j-2*n)^perinv +1 +2*n   else ((j-2*n)^perinv -1)^per : j in [2*n+1..4*n] ];
res cat:= resbis;
return doubleG!res;
end function;
numberofcycles := function(per);   ess :=   CycleStructure(per); return &+[ess[i,2]: i in [1..#ess]]; end function;
supernumberofcycles := function(per); return  numberofcycles(psifct(per)) ; end function;
// result given as a list genuslist (n+2-2g)^^multiplicity where g is the genus
// Case UO
dbl, dblsize := DoubleCosetRepresentatives(G,H,Cbetarev); #dblsize;
genuslist := {* supernumberofcycles(beta^(dbl[j]^(-1))) : j in [1..#dblsize] *}; genuslist;
quit;
// Robert Coquereaux, Nov 23 2015

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

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

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 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.

Cf. A008986, A008987, A008988, A008989, A007756, A264759, A264760.

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

a(15)-a(16) using J. Betrema's program added by Andrey Zabolotskiy, Aug 24 2023

a(17)-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

A268561 Number of bicolored immersions of unoriented circle into oriented sphere with n double points.

Original entry on oeis.org

1, 2, 3, 12, 37, 198, 1143, 7658, 54559, 413086, 3251240, 26484550, 221714164, 1900156948, 16613990484, 147816887598, 1335367515821

Offset: 0

N. J. A. Sloane, Mar 02 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 2, Table 5, and Table 9, line UOc.

Cf. A008986, A008987.

a(2*n) = A008986(2*n), a(2*n+1) = 2 * A008987(2*n+1). - Andrey Zabolotskiy, Jun 05 2024

Name clarified and a(0), a(11)-a(16) added using formula by Andrey Zabolotskiy, Jun 08 2024

A268569 Number of immersions of unoriented circle into oriented sphere with n double points and no simple loop.

Original entry on oeis.org

0, 0, 1, 1, 2, 6, 19, 74, 320, 1469

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. A008987, A264760, A268568, A268570, A268571, A268572.

Name clarified by Andrey Zabolotskiy, Jun 09 2024

cp's OEIS Frontend

A008986 Number of immersions of oriented circle into oriented sphere with n double points.

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A008988 Number of immersions of oriented circle into unoriented sphere with n double points.

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Comments

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Crossrefs

Programs

Magma

Extensions

A008989 Number of immersions of an unoriented circle into the unoriented sphere with n double points.

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Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Extensions

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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C

Extensions

A260848 Triangle read by rows: T(n,g) = number of general immersions of a circle with n crossings in a surface of arbitrary genus g (the circle is not oriented, the surface is oriented).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

C

Extensions

A264761 Number of irreducible indecomposable spherical curves with n crossings (only ordinary double points), the circle is oriented, the sphere is oriented (OO case).

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Author

Keywords

Comments

Links

Crossrefs

Programs

C

Extensions

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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