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 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. 6
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. Row g=0 is A008986 starting with n = 1. 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 LINKS Table of n, a(n) for n=1..45. 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 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 PROG (Magma) /* 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 CROSSREFS The sum over all genera g for a fixed number n of crossings is given by sequence A260296. Cf. A008986, A260285, A260848, A260914. Sequence in context: A095069 A184061 A222057 * A242499 A354622 A173020 Adjacent sequences: A260282 A260283 A260284 * A260286 A260287 A260288 KEYWORD nonn,tabl,hard AUTHOR Robert Coquereaux, Jul 22 2015 STATUS approved

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Last modified September 12 05:07 EDT 2024. Contains 375842 sequences. (Running on oeis4.)