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K3Carpets :: carpetBettiTables

carpetBettiTables -- compute the Betti tables of a carpet of given genus and Clifford index over all prime fields

Synopsis

Description

We compute the equation and nonminimal resolution F of the carpet of type (a,b) where a ≥b over a larger finite prime field, lift the complex to the integers, which is possible since the coefficients are small. Finally we study the nonminimal strands over ZZ by computing the Smith normal form. The resulting data allow us to compute the Betti tables for arbitrary primes.

i1 : a=5,b=5

o1 = (5, 5)

o1 : Sequence
i2 : h=carpetBettiTables(a,b)
     -- 0.00177743 seconds elapsed
     -- 0.00544697 seconds elapsed
     -- 0.0244545 seconds elapsed
     -- 0.00971853 seconds elapsed
     -- 0.0030625 seconds elapsed

                           0  1   2   3   4   5   6   7  8 9
o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 315 288   .   .   .  . .
                        2: .  .   .   .   . 288 315 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1
                           0  1   2   3   4   5   6   7  8 9
               2 => total: 1 36 167 370 476 476 370 167 36 1
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 322 336 140  48   7  . .
                        2: .  .   7  48 140 336 322 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1
                           0  1   2   3   4   5   6   7  8 9
               3 => total: 1 36 160 315 302 302 315 160 36 1
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 315 288  14   .   .  . .
                        2: .  .   .   .  14 288 315 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1

o2 : HashTable
i3 : T= carpetBettiTable(h,3)

            0  1   2   3   4   5   6   7  8 9
o3 = total: 1 36 160 315 302 302 315 160 36 1
         0: 1  .   .   .   .   .   .   .  . .
         1: . 36 160 315 288  14   .   .  . .
         2: .  .   .   .  14 288 315 160 36 .
         3: .  .   .   .   .   .   .   .  . 1

o3 : BettiTally
i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);

              ZZ
o4 : Ideal of --[x , x , x , x , x , x , y , y , y , y , y , y ]
               3  0   1   2   3   4   5   0   1   2   3   4   5
i5 : elapsedTime T'=minimalBetti J
     -- 0.30942 seconds elapsed

            0  1   2   3   4   5   6   7  8 9
o5 = total: 1 36 160 315 302 302 315 160 36 1
         0: 1  .   .   .   .   .   .   .  . .
         1: . 36 160 315 288  14   .   .  . .
         2: .  .   .   .  14 288 315 160 36 .
         3: .  .   .   .   .   .   .   .  . 1

o5 : BettiTally
i6 : T-T'

            0 1 2 3 4 5 6 7 8 9
o6 = total: . . . . . . . . . .
         1: . . . . . . . . . .
         2: . . . . . . . . . .
         3: . . . . . . . . . .

o6 : BettiTally
i7 : elapsedTime h=carpetBettiTables(6,6);
     -- 0.00381003 seconds elapsed
     -- 0.0458444 seconds elapsed
     -- 0.13603 seconds elapsed
     -- 1.38229 seconds elapsed
     -- 0.489425 seconds elapsed
     -- 0.0394654 seconds elapsed
     -- 0.00572176 seconds elapsed
     -- 9.27678 seconds elapsed
i8 : keys h

o8 = {0, 2, 3, 5}

o8 : List
i9 : carpetBettiTable(h,7)

            0  1   2   3    4    5    6    7   8   9 10 11
o9 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
         0: 1  .   .   .    .    .    .    .   .   .  .  .
         1: . 55 320 891 1408 1155    .    .   .   .  .  .
         2: .  .   .   .    .    . 1155 1408 891 320 55  .
         3: .  .   .   .    .    .    .    .   .   .  .  1

o9 : BettiTally
i10 : carpetBettiTable(h,5)

             0  1   2   3    4    5    6    7   8   9 10 11
o10 = total: 1 55 320 891 1408 1275 1275 1408 891 320 55  1
          0: 1  .   .   .    .    .    .    .   .   .  .  .
          1: . 55 320 891 1408 1155  120    .   .   .  .  .
          2: .  .   .   .    .  120 1155 1408 891 320 55  .
          3: .  .   .   .    .    .    .    .   .   .  .  1

o10 : BettiTally

See also

Ways to use carpetBettiTables :