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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 $a \ge b$ 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)
 -- 0.0738407 seconds elapsed
 -- 0.174564 seconds elapsed
 -- 0.907733 seconds elapsed
 -- 0.320815 seconds elapsed
 -- 0.149088 seconds elapsed
 -- 6.49646 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 , y ..y ]
               3  0   5   0   5
i4 : elapsedTime T'=minimalBetti J
 -- 10.1797 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);
 -- 0.154943 seconds elapsed
 -- 0.469431 seconds elapsed
 -- 2.80291 seconds elapsed
 -- 27.5589 seconds elapsed
 -- 12.7775 seconds elapsed
 -- 1.13502 seconds elapsed
 -- 0.194 seconds elapsed
 -- 100.86 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 :

For the programmer

The object carpetBettiTable is a method function.