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

## Synopsis

• Usage:
h = carpetBettiTable(a,b,p)
carpetBettiTable(h,p)
• Inputs:
• a, an integer, the larger value, i.e a>=b
• b, an integer, the desired clifford index
• h, , of carpet Betti tables for various characteristics of the ground field
• p, an integer, the characteristic of the ground field
• Outputs:
• , the betti Table of a carpet of the given ground field

## 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) -- 0.00362144 seconds elapsed -- 0.0105768 seconds elapsed -- 0.0937814 seconds elapsed -- 0.0165722 seconds elapsed -- 0.00613265 seconds elapsed -- 0.453701 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 -- 0.461066 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); -- 0.00787529 seconds elapsed -- 0.030215 seconds elapsed -- 0.253969 seconds elapsed -- 2.7472 seconds elapsed -- 0.789109 seconds elapsed -- 0.0678934 seconds elapsed -- 0.00979759 seconds elapsed -- 12.3529 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