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

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)
 -- .00180728s elapsed
 -- .00466778s elapsed
 -- .0151865s elapsed
 -- .00740115s elapsed
 -- .00255223s elapsed
 -- .242085s 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
 -- .132075s 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);
 -- .00334061s elapsed
 -- .0119189s elapsed
 -- .0948489s elapsed
 -- .668183s elapsed
 -- .307193s elapsed
 -- .0248758s elapsed
 -- .00493733s elapsed
 -- 3.79821s 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:

  • carpetBettiTable(HashTable,ZZ)
  • carpetBettiTable(ZZ,ZZ,ZZ)

For the programmer

The object carpetBettiTable is a method function.


The source of this document is in K3Carpets.m2:1498:0.