A008986
Number of immersions of oriented circle into oriented sphere with n double points.
Original entry on oeis.org
1, 1, 3, 9, 37, 182, 1143, 7553, 54559, 412306, 3251240, 26478264, 221714164, 1900103364, 16613990484, 147816411921, 1335367515821
Offset: 0
- V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994, p. 18.
A008987
Number of immersions of an unoriented circle into the oriented sphere with n double points.
Original entry on oeis.org
1, 1, 2, 6, 21, 99, 588, 3829, 27404, 206543, 1626638, 13242275, 110865868, 950078474, 8307074080, 73908443799, 667684486429
Offset: 0
- V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994, p. 18.
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.
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
- 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.
Comment on link to plantri modified by
Brendan McKay, Mar 25 2024
A260885
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 oriented, the surface is unoriented).
Original entry on oeis.org
1, 2, 1, 6, 6, 2, 21, 62, 37, 0, 97, 559, 788, 112, 0, 579, 5614, 14558, 7223, 0, 0, 3812, 56526, 246331, 277407, 34748, 0, 0, 27328, 580860, 3900740, 8179658, 3534594, 0, 0, 0, 206410, 6020736, 58842028, 203974134, 198559566, 22524176, 0, 0, 0
Offset: 1
The transposed triangle starts:
1 2 6 21 97 579 3812 27328 206410
1 6 62 559 5614 56526 580860 6020736
2 37 788 14558 246331 3900740 58842028
0 112 7223 277407 8179658 203974134
0 0 34748 3534594 198559566
0 0 0 22524176
0 0 0
0 0
0
- Robert 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
-
/* 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 OU
dbl, dblsize := DoubleCosetRepresentatives(G,Hcycrho,Cbeta); #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
- 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.
A268570
Number of immersions of oriented circle into unoriented sphere with n double points and no simple loop.
Original entry on oeis.org
0, 0, 1, 1, 2, 5, 18, 70, 313, 1440
Offset: 1
A268565
Number of achiral immersions of oriented circle into oriented sphere with 2n double points.
Original entry on oeis.org
1, 1, 5, 15, 97, 592, 4488, 35314, 295379
Offset: 0
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