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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). 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 A008988 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

LINKS

Table of n, a(n) for n=1..45.

R. Coquereaux, J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: http://dx.doi.org/10.1142/S0218216516500474

EXAMPLE

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

PROG

(MAGMA) /* 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<G|[H, cycrho]>;  // 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<G|[Cbeta, cycbeta]>;

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

CROSSREFS

Cf. A008988. The sum over all genera g for a fixed number n of crossings is given by sequence A260887. Cf. A260885, A260848, A260914.

Sequence in context: A232433 A271881 A182729 * A075181 A052121 A193895

Adjacent sequences:  A260882 A260883 A260884 * A260886 A260887 A260888

KEYWORD

nonn,tabl,hard

AUTHOR

Robert Coquereaux, Aug 02 2015

STATUS

approved

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Last modified October 28 17:59 EDT 2021. Contains 348329 sequences. (Running on oeis4.)