next | previous | forward | backward | up | top | index | toc | packages | Macaulay2 website
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)
 -- 0.00201984 seconds elapsed
 -- 0.00550072 seconds elapsed
 -- 0.0231074 seconds elapsed
 -- 0.0330133 seconds elapsed
 -- 0.00334974 seconds elapsed
 -- 0.226032 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
 -- 0.191643 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);
 -- 0.00420611 seconds elapsed
 -- 0.0174698 seconds elapsed
 -- 0.142873 seconds elapsed
 -- 1.28557 seconds elapsed
 -- 0.416117 seconds elapsed
 -- 0.0381035 seconds elapsed
 -- 0.00638366 seconds elapsed
 -- 4.38668 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.