A008989
Number of immersions of an unoriented circle into the unoriented sphere with n double points.
Original entry on oeis.org
1, 1, 2, 6, 19, 76, 376, 2194, 14614, 106421, 823832, 6657811, 55557329, 475475046, 4155030702, 36959470662, 333860366236
Offset: 0
G.f. = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 76*x^5 + 376*x^6 + 2194*x^7 + ...
- V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994, p. 18.
- J. Cantarella, H. Chapman, and M. Mastin, Knot Probabilities in Random Diagrams, arXiv preprint arXiv:1512.05749 [math.GT], 2015. Also Journal of Physics A: Mathematical and Theoretical, Vol. 49, No. 40 (2016), DOI: 10.1088/1751-8113/49/40/405001
- 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), 10.1142/S0218216516500474.
- Guy Valette, A Classification of Spherical Curves Based on Gauss Diagrams, Arnold Math J. (2016) 2:383-405.
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.
Original entry on oeis.org
1, 3, 1, 9, 11, 2, 37, 113, 68, 0, 182, 1102, 1528, 216, 0, 1143, 11114, 28947, 14336, 0, 0, 7553, 112846, 491767, 554096, 69264, 0, 0, 54559, 1160532, 7798139, 16354210, 7066668, 0, 0, 0, 412306, 12038974, 117668914, 407921820, 397094352, 45043200, 0, 0, 0
Offset: 1
The transposed triangle starts:
1 3 9 37 182 1143 7553 54559 412306
1 11 113 1102 11114 112846 1160532 12038974
2 68 1528 28947 491767 7798139 117668914
0 216 14336 554096 16354210 407921820
0 0 69264 7066668 397094352
0 0 0 45043200
0 0 0
0 0
- 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
-
/* 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 OO
dbl, dblsize := DoubleCosetRepresentatives(G,H,Cbeta); #dblsize;
genuslist := {* supernumberofcycles(beta^(dbl[j]^(-1))) : j in [1..#dblsize] *}; genuslist;
quit;
// Robert Coquereaux, Nov 23 2015
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
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
The sum over all genera g for a fixed number n of crossings is given by sequence
A260847.
-
/* 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
A260885
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 oriented, the surface is unoriented).
Original entry on oeis.org
1, 2, 1, 6, 6, 2, 21, 62, 37, 0, 97, 559, 788, 112, 0, 579, 5614, 14558, 7223, 0, 0, 3812, 56526, 246331, 277407, 34748, 0, 0, 27328, 580860, 3900740, 8179658, 3534594, 0, 0, 0, 206410, 6020736, 58842028, 203974134, 198559566, 22524176, 0, 0, 0
Offset: 1
The transposed triangle starts:
1 2 6 21 97 579 3812 27328 206410
1 6 62 559 5614 56526 580860 6020736
2 37 788 14558 246331 3900740 58842028
0 112 7223 277407 8179658 203974134
0 0 34748 3534594 198559566
0 0 0 22524176
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: http://dx.doi.org/10.1142/S0218216516500474
-
/* 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 OU
dbl, dblsize := DoubleCosetRepresentatives(G,Hcycrho,Cbeta); #dblsize;
genuslist := {* supernumberofcycles(beta^(dbl[j]^(-1))) : j in [1..#dblsize] *}; genuslist;
quit;
// Robert Coquereaux, Nov 23 2015
A260912
Sum over the genera g of the number of immersions of an unoriented circle with n crossings in an unoriented surface of genus g.
Original entry on oeis.org
1, 3, 12, 86, 894, 14715, 313364, 8139398, 245237925, 8382002270, 319994166042, 13492740284184, 622738642693202, 31225868370080949, 1690360086869176780, 98252177808632109236, 6103194081506193327048, 403488941845715112039425, 28286698447226523233226110, 2096044354918091666701275248
Offset: 1
- 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
-
/* 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;
UUfull := 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); bool, rever := IsConjugate(G,beta,beta^(-1)); cycbeta := PermutationGroup< 2*n |{rever}>;
Cbetarev := sub;
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,Cbetarev); end function;
[UUfull(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
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
...
-
/* 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;
Showing 1-6 of 6 results.
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