A260285 Triangle read by rows: T(n,g) = number of general immersions of a circle with n crossings in a surface of arbitrary genus g, in the case that the circle is oriented and the surface is oriented.

1, 3, 1, 9, 11, 2, 37, 113, 68, 0, 182, 1102, 1528, 216, 0, 1143, 11114, 28947, 14336, 0, 0, 7553, 112846, 491767, 554096, 69264, 0, 0, 54559, 1160532, 7798139, 16354210, 7066668, 0, 0, 0, 412306, 12038974, 117668914, 407921820, 397094352, 45043200, 0, 0, 0

Offset: 1

Author

Robert Coquereaux, Jul 22 2015

Keywords

• nonn
• tabl
• hard

Comments

When transposed, displayed as an upper right triangle, and read by columns, 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 A008986 starting with n = 1.

For g > 0 the immersions are understood up to stable geotopy equivalence (listed curves cannot be immersed in a surface of smaller genus). - Robert Coquereaux, Nov 23 2015

Examples

			The transposed triangle starts:
  1  3   9   37   182    1143      7553      54559            412306
     1  11  113  1102   11114    112846    1160532          12038974
         2   68  1528   28947    491767    7798139         117668914
              0   216   14336    554096   16354210         407921820
                    0      0      69264    7066668         397094352
                           0          0         0           45043200
                                      0         0                  0
                                                0                  0
		

Links

• 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

Crossrefs

The sum over all genera g for a fixed number n of crossings is given by sequence A260296. Cf. A008986, A260285, A260848, A260914.

Programs

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;  // 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 OO
dbl, dblsize := DoubleCosetRepresentatives(G,H,Cbeta); #dblsize;
genuslist := {* supernumberofcycles(beta^(dbl[j]^(-1))) : j in [1..#dblsize] *}; genuslist;
quit;
// Robert Coquereaux, Nov 23 2015