cp's OEIS Frontend

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.

%I A260285 #31 Nov 09 2024 19:34:20
%S A260285 1,3,1,9,11,2,37,113,68,0,182,1102,1528,216,0,1143,11114,28947,14336,
%T A260285 0,0,7553,112846,491767,554096,69264,0,0,54559,1160532,7798139,
%U A260285 16354210,7066668,0,0,0,412306,12038974,117668914,407921820,397094352,45043200,0,0,0
%N 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.
%C A260285 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.
%C A260285 Row g=0 is A008986 starting with n = 1.
%C A260285 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
%H A260285 R. Coquereaux, J.-B. Zuber, <a href="http://arxiv.org/abs/1507.03163">Maps, immersions and permutations</a>, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: http://dx.doi.org/10.1142/S0218216516500474
%e A260285 The transposed triangle starts:
%e A260285   1  3   9   37   182    1143      7553      54559            412306
%e A260285      1  11  113  1102   11114    112846    1160532          12038974
%e A260285          2   68  1528   28947    491767    7798139         117668914
%e A260285               0   216   14336    554096   16354210         407921820
%e A260285                     0      0      69264    7066668         397094352
%e A260285                            0          0         0           45043200
%e A260285                                       0         0                  0
%e A260285                                                 0                  0
%o A260285 (Magma) /* Example n := 6 */
%o A260285 n:=6;
%o A260285 n; // n: number of crossings
%o A260285 G:=Sym(2*n);
%o A260285 doubleG := Sym(4*n);
%o A260285 genH:={};
%o A260285 for j in [1..(n-1)] do v := G!(1,2*j+1)(2, 2*j+2); Include(~genH,v) ; end for;
%o A260285 H := PermutationGroup< 2*n |genH>; //  The H=S(n) subgroup of S(2n)
%o A260285 cardH:=#H;
%o A260285 cardH;
%o A260285 rho:=Identity(G); for j in [0..(n-1)] do v := G!(2*j+1, 2*j+2) ; rho := rho*v ; end for;
%o A260285 cycrho := PermutationGroup< 2*n |{rho}>; // The cyclic subgroup Z2 generated by rho (mirroring)
%o A260285 Hcycrho:=sub<G|[H,cycrho]>;  // The subgroup generated by H and cycrho
%o A260285 cardZp:= Factorial(2*n-1);
%o A260285 beta:=G!Append([2..2*n],1); // A typical circular permutation
%o A260285 Cbeta:=Centralizer(G,beta);
%o A260285 bool, rever := IsConjugate(G,beta,beta^(-1));
%o A260285 cycbeta := PermutationGroup< 2*n |{rever}>;
%o A260285 Cbetarev := sub<G|[Cbeta,cycbeta]>;
%o A260285 psifct := function(per);
%o A260285 perinv:=per^(-1);
%o A260285 res:= [IsOdd(j) select (j+1)^per  else j-1 + 2*n : j in [1..2*n] ];
%o A260285 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] ];
%o A260285 res cat:= resbis;
%o A260285 return doubleG!res;
%o A260285 end function;
%o A260285 numberofcycles := function(per);   ess :=   CycleStructure(per); return &+[ess[i,2]: i in [1..#ess]]; end function;
%o A260285 supernumberofcycles := function(per); return  numberofcycles(psifct(per)) ; end function;
%o A260285 // result given as a list genuslist (n+2-2g)^^multiplicity where g is the genus
%o A260285 //case OO
%o A260285 dbl, dblsize := DoubleCosetRepresentatives(G,H,Cbeta); #dblsize;
%o A260285 genuslist := {* supernumberofcycles(beta^(dbl[j]^(-1))) : j in [1..#dblsize] *}; genuslist;
%o A260285 quit;
%o A260285 // _Robert Coquereaux_, Nov 23 2015
%Y A260285 The sum over all genera g for a fixed number n of crossings is given by sequence A260296. Cf. A008986, A260285, A260848, A260914.
%K A260285 nonn,tabl,hard
%O A260285 1,2
%A A260285 _Robert Coquereaux_, Jul 22 2015