A260885 -id:A260885 - cp's OEIS frontend

A008988 Number of immersions of oriented circle into unoriented sphere with n double points.

1, 1, 2, 6, 21, 97, 579, 3812, 27328, 206410, 1625916, 13241177, 110859326, 950069179, 8307012899, 73908363060, 667683905600

Offset: 0

N. J. A. Sloane

a(10) = 1625916 by sampling method (to be confirmed, cf. ref. arXiv:1507.03163). - Robert Coquereaux, Nov 23 2015

a(10) = 1625916 confirmed. - Brendan McKay, Mar 29 2024

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994, p. 18.
Guy Valette, A Classification of Spherical Curves Based on Gauss Diagrams, Arnold Math J. (2016) 2:383-405, DOI 10.1007/s40598-016-0049-3.

R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also DOI, J. Knot Theory Ramifications 25, 1650047 (2016).

Cf. A008986, A008987, A008989. First line of triangle A260885.

Magma
```
// see A260885
```

a(6)-a(7) from Guy Valette, Feb 09 2004
a(8)-a(9) from Robert Coquereaux and Jean-Bernard Zuber, Jul 21 2015
a(10) from same source added by N. J. A. Sloane, Mar 03 2016
a(11)-a(16) from Brendan McKay, Mar 29 2024

A260848 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 oriented).

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 21, 64, 36, 0, 99, 559, 772, 108, 0, 588, 5656, 14544, 7222, 0, 0, 3829, 56528, 246092, 277114, 34680, 0, 0, 27404, 581511, 3900698, 8180123, 3534038, 0, 0, 0, 206543, 6020787, 58838383, 203964446, 198551464, 22521600, 0, 0, 0

Offset: 1

Robert Coquereaux, Aug 01 2015

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.
Row g=0 is A008987 starting with n = 1.
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

			The transposed triangle starts:
1    2   6   21    99     588      3829      27404            206543
     1   6   64   559    5656     56528     581511           6020787
         1   36   772   14544    246092    3900698          58838383
              0   108    7222    277114    8180123         203964446
                    0      0      34680    3534038         198551464
                           0          0         0           22521600
                                      0         0                  0
                                                0                  0

R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv:1507.03163 [math.CO], 2015, Table 9.

The sum over all genera g for a fixed number n of crossings is given by sequence A260847.

Cf. A008987, A260285, A260885, A260914.

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

A260914 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).

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 19, 45, 22, 0, 76, 335, 427, 56, 0, 376, 3101, 7557, 3681, 0, 0, 2194, 29415, 124919, 139438, 17398, 0, 0, 14614, 295859, 1921246, 4098975, 1768704, 0, 0, 0, 106421, 3031458, 29479410, 102054037, 99304511, 11262088, 0, 0, 0

Offset: 1

Robert Coquereaux, Aug 04 2015

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

			The transposed triangle starts:
  1  2  6  19   76   376     2194     14614     106421
     1  5  45  335  3101    29415    295859    3031458
        1  22  427  7557   124919   1961246   29479410
            0   56  3681   139438   4098975  102054037
                 0    0     17398   1768704   99394511
                      0         0         0   11262088
                                0         0          0
                                          0          0
                                                     0

Robert Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: 10.1142/S0218216516500474

The sum over all genera g for a fixed number n of crossings is given by sequence A260912. Cf. A008989, A260285, A260848, A260885.

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

A260887 Sum over the genera g of the number of immersions of an oriented circle with n crossings in an unoriented surface of genus g.

Original entry on oeis.org

1, 3, 14, 120, 1556, 27974, 618824, 16223180, 490127050, 16761331644, 639969571892, 26985326408240, 1245476099801252, 62451726395242858, 3380720087847928728, 196504354827002278248, 12206388156005725243280, 806977883623811932432386, 56573396893818112613554940, 4192088709829783508863131872

Offset: 1

Robert Coquereaux, Aug 02 2015

nonn

a(n) is the sum over the n-th row of the triangle A260885.
a(n) is also the number of double cosets of H\G/K where G is the symmetric group S(2n), H is the subgroup generated by the centralizer of the circular permutation β = (1,2,3,...,2n) of G, K is a subgroup of G generated by the permutation ρ = (1,2)(3,4)...(2n-3,2n-2)(2n-1,2n), using cycle notation, and the subgroup (isomorphic with S(n)) that commutes with ρ and permutes odd resp. even integers among themselves.
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

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

Cf. A260296, A260847, A260885, A260912.

/* For all n */
nbofdblecos := function(G, H, K);
CG := Classes(G); nCG := #CG; oG := #G; CH := Classes(H); nCH := #CH; oH := #H; CK := Classes(K); nCK := #CK; oK := #K;
resH := []; for mu in [1..nCG] do  Gmurep := CG[mu][3]; Hmupositions := {j: j in [1..nCH]  | CycleStructure(CH[j][3]) eq CycleStructure(Gmurep)};
Hmugoodpositions := {j : j in Hmupositions | IsConjugate(G,CH[j][3], Gmurep) eq true}; bide := 0; for j in Hmugoodpositions do bide := bide + CH[j][2]; end for;
Append(~resH, bide); end for;
resK := []; for mu in [1..nCG] do  Gmurep := CG[mu][3]; Kmupositions := {j: j in [1..nCK]  | CycleStructure(CK[j][3]) eq CycleStructure(Gmurep)};
Kmugoodpositions := {j : j in Kmupositions | IsConjugate(G,CK[j][3], Gmurep) eq true}; bide := 0; for j in Kmugoodpositions do bide := bide + CK[j][2]; end for;
Append(~resK, bide); end for;
ndcl := 0; tot := 0; for mu in [1..nCG] do tot := tot + resH[mu]* resK[mu]/CG[mu][2]; end for;  ndcl:= tot *  oG/(oH * oK); return ndcl;
end function;
OUfull := function(n); G:=Sym(2*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>;
beta:=G!Append([2..2*n],1); Cbeta:=Centralizer(G,beta);
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}>;  Hcycrho:=sub;
return nbofdblecos(G,Hcycrho,Cbeta); end function;
[OUfull(n) : n in [1..10]]; //

A264755 Triangle T(n,g) read by rows: Partition of the set of (2n-1)! circular permutations on 2n elements according to the minimal genus g of the surface in which one can immerse the non-simple closed curves with n crossings determined by those permutations.

Original entry on oeis.org

1, 4, 2, 42, 66, 12, 780, 2652, 1608, 21552, 132240, 183168, 25920, 803760, 7984320, 20815440, 10313280

Offset: 1

Robert Coquereaux, Nov 23 2015

Each line of the triangle adds up to an odd factorial (2n-1)!. Example (line n=5): 21552 + 132240 + 183168 + 25920 = 362880 = 9!.

The lengths of the rows of the triangle do not strictly increase with n, the first lengths are (1,2,3,3,4,4,...).

			Taking n = 5 crossings and genus g=0, one obtains a subset of T(5, 0) = 21552 circular permutations of Sym(10) which correspond, in the OO case (the circle is oriented, the sphere is oriented), to the union 179 orbits of length 120=5!/1 and 3 orbits of length 24=5!/5 with respective centralizers of order 1 and 5 under the action of the symmetric group Sym(5) acting on this subset: 179*120 + 3*24 = 21552. The total number of orbits 179 + 3 = 182 = A008986(5) = A260285(5, 0) is the number of immersed spherical curves (g=0) with 5 crossings, in the OO case. The next entry, T(5, 1) = 132240, gives the number of circular permutations that describe immersed closed curves in a torus (g=1), with n=5 crossings, up to stable geotopy; the number of such closed curves in the OO case is 1102 = A260285(5, 1).
Triangle begins:
  1
  4 2
  42 66 12
  780 2652 1608
  21552 132240 183168 25920
  803760 7984320 20815440 10313280
  ...

R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: 10.1142/S0218216516500474

Cf. A008986, A260296.

Cf. A260285, A260848, A260885, A260914, A268567 (g=0).

/* Example: line n=5 of the triangle */
n:=5;
G:=Sym(2*n);
CG := Classes(G);
pos:= [j: j in [1..#CG]  | CycleStructure(CG[j][3]) eq [<2*n,1>]][1];
circularpermutations:=Class(G,CG[pos][3]); //circularpermutations
doubleG := Sym(4*n);
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;
{* supernumberofcycles(x) : x in circularpermutations  *};
quit;

cp's OEIS Frontend

A008988 Number of immersions of oriented circle into unoriented sphere with n double points.

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A260848 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 oriented).

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A260914 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).

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A260887 Sum over the genera g of the number of immersions of an oriented circle with n crossings in an unoriented surface of genus g.

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A264755 Triangle T(n,g) read by rows: Partition of the set of (2n-1)! circular permutations on 2n elements according to the minimal genus g of the surface in which one can immerse the non-simple closed curves with n crossings determined by those permutations.

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