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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)
     -- 0.00118425 seconds elapsed
     -- 0.00162756 seconds elapsed
     -- 0.00189763 seconds elapsed
     -- 0.00179268 seconds elapsed
     -- 0.00138618 seconds elapsed

                           0  1   2   3   4   5   6   7  8 9
o2 = HashTable{0 => total: 1 36 209 516 786 786 516 209 36 1}
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 342 436 350 174  49  . .
                        2: .  .  49 174 350 436 342 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1

o2 : HashTable
i3 : T= carpetBettiTable(h,3)

            0  1   2   3   4   5   6   7  8 9
o3 = total: 1 36 209 516 786 786 516 209 36 1
         0: 1  .   .   .   .   .   .   .  . .
         1: . 36 160 342 436 350 174  49  . .
         2: .  .  49 174 350 436 342 160 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
     -- 0.227086 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: . . 49 201 484 484 201 49 . .
         1: . .  .  27 148 336 174 49 . .
         2: . . 49 174 336 148  27  . . .
         3: . .  .   .   .   .   .  . . .

o6 : BettiTally
i7 : elapsedTime h=carpetBettiTables(6,6);
     -- 0.00239531 seconds elapsed
     -- 0.00439149 seconds elapsed
     -- 0.00656188 seconds elapsed
     -- 0.00737626 seconds elapsed
     -- 0.00685538 seconds elapsed
     -- 0.00522131 seconds elapsed
     -- 0.00275713 seconds elapsed
     -- 68.6065 seconds elapsed
i8 : keys h

o8 = {0}

o8 : List
i9 : carpetBettiTable(h,7)

            0  1   2    3    4    5    6    7    8   9 10 11
o9 = total: 1 55 401 1298 2675 3788 3788 2675 1298 401 55  1
         0: 1  .   .    .    .    .    .    .    .   .  .  .
         1: . 55 320  930 1688 2060 1728  987  368  81  .  .
         2: .  .  81  368  987 1728 2060 1688  930 320 55  .
         3: .  .   .    .    .    .    .    .    .   .  .  1

o9 : BettiTally
i10 : carpetBettiTable(h,5)

             0  1   2    3    4    5    6    7    8   9 10 11
o10 = total: 1 55 401 1298 2675 3788 3788 2675 1298 401 55  1
          0: 1  .   .    .    .    .    .    .    .   .  .  .
          1: . 55 320  930 1688 2060 1728  987  368  81  .  .
          2: .  .  81  368  987 1728 2060 1688  930 320 55  .
          3: .  .   .    .    .    .    .    .    .   .  .  1

o10 : BettiTally

See also

Ways to use carpetBettiTables :