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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 ≥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)
     -- 0.001188 seconds elapsed
     -- 0.00164194 seconds elapsed
     -- 0.00192062 seconds elapsed
     -- 0.00184112 seconds elapsed
     -- 0.00136806 seconds elapsed
     -- 0.245663 seconds elapsed

            0  1   2   3   4   5   6   7  8 9
o2 = 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 : BettiTally
i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);

              ZZ
o3 : 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
i4 : elapsedTime T'=minimalBetti J
     -- 0.223932 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: . . 49 201 484 484 201 49 . .
         1: . .  .  27 148 336 174 49 . .
         2: . . 49 174 336 148  27  . . .
         3: . .  .   .   .   .   .  . . .

o5 : BettiTally
i6 : elapsedTime h=carpetBettiTables(6,6);
     -- 0.002427 seconds elapsed
     -- 0.004433 seconds elapsed
     -- 0.00649469 seconds elapsed
     -- 0.00738594 seconds elapsed
     -- 0.00694956 seconds elapsed
     -- 0.005222 seconds elapsed
     -- 0.00273112 seconds elapsed
     -- 68.2796 seconds elapsed
i7 : carpetBettiTable(h,7)

            0  1   2    3    4    5    6    7    8   9 10 11
o7 = 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

o7 : BettiTally
i8 : carpetBettiTable(h,5)

            0  1   2    3    4    5    6    7    8   9 10 11
o8 = 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

o8 : BettiTally

See also

Ways to use carpetBettiTable :