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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.
6

%I #31 Nov 09 2024 19:34:20

%S 1,3,1,9,11,2,37,113,68,0,182,1102,1528,216,0,1143,11114,28947,14336,

%T 0,0,7553,112846,491767,554096,69264,0,0,54559,1160532,7798139,

%U 16354210,7066668,0,0,0,412306,12038974,117668914,407921820,397094352,45043200,0,0,0

%N 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 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 Row g=0 is A008986 starting with n = 1.

%C 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 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 The transposed triangle starts:

%e 1 3 9 37 182 1143 7553 54559 412306

%e 1 11 113 1102 11114 112846 1160532 12038974

%e 2 68 1528 28947 491767 7798139 117668914

%e 0 216 14336 554096 16354210 407921820

%e 0 0 69264 7066668 397094352

%e 0 0 0 45043200

%e 0 0 0

%e 0 0

%o (Magma) /* Example n := 6 */

%o n:=6;

%o n; // n: number of crossings

%o G:=Sym(2*n);

%o doubleG := Sym(4*n);

%o genH:={};

%o for j in [1..(n-1)] do v := G!(1,2*j+1)(2, 2*j+2); Include(~genH,v) ; end for;

%o H := PermutationGroup< 2*n |genH>; // The H=S(n) subgroup of S(2n)

%o cardH:=#H;

%o cardH;

%o rho:=Identity(G); for j in [0..(n-1)] do v := G!(2*j+1, 2*j+2) ; rho := rho*v ; end for;

%o cycrho := PermutationGroup< 2*n |{rho}>; // The cyclic subgroup Z2 generated by rho (mirroring)

%o Hcycrho:=sub<G|[H,cycrho]>; // The subgroup generated by H and cycrho

%o cardZp:= Factorial(2*n-1);

%o beta:=G!Append([2..2*n],1); // A typical circular permutation

%o Cbeta:=Centralizer(G,beta);

%o bool, rever := IsConjugate(G,beta,beta^(-1));

%o cycbeta := PermutationGroup< 2*n |{rever}>;

%o Cbetarev := sub<G|[Cbeta,cycbeta]>;

%o psifct := function(per);

%o perinv:=per^(-1);

%o res:= [IsOdd(j) select (j+1)^per else j-1 + 2*n : j in [1..2*n] ];

%o 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 res cat:= resbis;

%o return doubleG!res;

%o end function;

%o numberofcycles := function(per); ess := CycleStructure(per); return &+[ess[i,2]: i in [1..#ess]]; end function;

%o supernumberofcycles := function(per); return numberofcycles(psifct(per)) ; end function;

%o // result given as a list genuslist (n+2-2g)^^multiplicity where g is the genus

%o //case OO

%o dbl, dblsize := DoubleCosetRepresentatives(G,H,Cbeta); #dblsize;

%o genuslist := {* supernumberofcycles(beta^(dbl[j]^(-1))) : j in [1..#dblsize] *}; genuslist;

%o quit;

%o // _Robert Coquereaux_, Nov 23 2015

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

%K nonn,tabl,hard,changed

%O 1,2

%A _Robert Coquereaux_, Jul 22 2015