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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 (the circle is not oriented, the surface is not oriented).
6

%I #21 Sep 08 2022 08:46:13

%S 1,2,1,6,5,1,19,45,22,0,76,335,427,56,0,376,3101,7557,3681,0,0,2194,

%T 29415,124919,139438,17398,0,0,14614,295859,1921246,4098975,1768704,0,

%U 0,0,106421,3031458,29479410,102054037,99304511,11262088,0,0,0

%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 (the circle is not oriented, the surface is not oriented).

%C 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.

%C Row g=0 is A008989 starting with n = 1.

%C 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

%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 2 6 19 76 376 2194 14614 106421

%e 1 5 45 335 3101 29415 295859 3031458

%e 1 22 427 7557 124919 1961246 29479410

%e 0 56 3681 139438 4098975 102054037

%e 0 0 17398 1768704 99394511

%e 0 0 0 11262088

%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 UU

%o dbl, dblsize := DoubleCosetRepresentatives(G,Hcycrho,Cbetarev); #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 A260912. Cf. A008989, A260285, A260848, A260885.

%K nonn,hard,tabl

%O 1,2

%A _Robert Coquereaux_, Aug 04 2015