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K3Carpets :: carpetBettiTable

carpetBettiTable -- compute the Betti tables of a carpet of given genus and Clifford index over a prime field of characteristic p

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 : elapsedTime T=carpetBettiTable(a,b,3)
     -- 0.00249669 seconds elapsed
     -- 0.00961185 seconds elapsed
     -- 0.0457215 seconds elapsed
     -- 0.0170099 seconds elapsed
     -- 0.0046874 seconds elapsed
     -- 0.37904 seconds elapsed

            0  1   2   3   4   5   6   7  8 9
o2 = 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 : BettiTally
i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);

              ZZ
o3 : 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
i4 : elapsedTime T'=minimalBetti J
     -- 0.299679 seconds elapsed

            0  1   2   3   4   5   6   7  8 9
o4 = 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

o4 : BettiTally
i5 : T-T'

            0 1 2 3 4 5 6 7 8 9
o5 = total: . . . . . . . . . .
         1: . . . . . . . . . .
         2: . . . . . . . . . .
         3: . . . . . . . . . .

o5 : BettiTally
i6 : elapsedTime h=carpetBettiTables(6,6);
     -- 0.00522134 seconds elapsed
     -- 0.0276359 seconds elapsed
     -- 0.206582 seconds elapsed
     -- -1.88385 seconds elapsed
     -- 0.600987 seconds elapsed
     -- 0.0665801 seconds elapsed
     -- 0.00867086 seconds elapsed
     -- -0.366195 seconds elapsed
i7 : carpetBettiTable(h,7)

            0  1   2   3    4    5    6    7   8   9 10 11
o7 = 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

o7 : BettiTally
i8 : carpetBettiTable(h,5)

            0  1   2   3    4    5    6    7   8   9 10 11
o8 = 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

o8 : BettiTally

See also

Ways to use carpetBettiTable :