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%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
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