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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)
     -- 0.00213563 seconds elapsed
     -- 0.0081956 seconds elapsed
     -- 0.0409143 seconds elapsed
     -- 0.0136447 seconds elapsed
     -- 0.0036008 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
     -- 0.217245 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);
     -- 0.00492677 seconds elapsed
     -- 0.026736 seconds elapsed
     -- 0.204776 seconds elapsed
     -- -2.06485 seconds elapsed
     -- 0.716984 seconds elapsed
     -- 0.0626426 seconds elapsed
     -- 0.00818945 seconds elapsed
     -- -0.804169 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 :