# analyzeStrand -- analyze the (a+1)-st constant strand of F over ZZ

## Synopsis

• Usage:
L = analyzeStrand(F,a)
• Inputs:
• F, , the nonminimal resolution of a carpet over large prime
• a, an integer, a+1 is the degree of the strand
• Outputs:
• L, a list, the diagonal entries of the Smith normal form of a the cruital matrix

## Description

Starting from a nonminimal resolution F of the carpet over a larger finite prime field, we lift the complex to the integers, and compute the diagonal entries of the Smith normal form. The critical constrand strand for a carpet of type (a,b) with a>=b is the a+1-st strand. Green's conjecture for carpet says that the map has maximal rank over QQ.

 i1 : a=5,b=5 o1 = (5, 5) o1 : Sequence i2 : I = carpet(a,b); ZZ o2 : Ideal of -----[x ..x , y ..y ] 32003 0 5 0 5 i3 : F = res(I, FastNonminimal => true) ZZ 1 ZZ 36 ZZ 187 ZZ 491 ZZ 793 ZZ 833 ZZ 573 ZZ 250 ZZ 63 ZZ 7 o3 = (-----[x ..x , y ..y ]) <-- (-----[x ..x , y ..y ]) <-- (-----[x ..x , y ..y ]) <-- (-----[x ..x , y ..y ]) <-- (-----[x ..x , y ..y ]) <-- (-----[x ..x , y ..y ]) <-- (-----[x ..x , y ..y ]) <-- (-----[x ..x , y ..y ]) <-- (-----[x ..x , y ..y ]) <-- (-----[x ..x , y ..y ]) <-- 0 32003 0 5 0 5 32003 0 5 0 5 32003 0 5 0 5 32003 0 5 0 5 32003 0 5 0 5 32003 0 5 0 5 32003 0 5 0 5 32003 0 5 0 5 32003 0 5 0 5 32003 0 5 0 5 10 0 1 2 3 4 5 6 7 8 9 o3 : ChainComplex i4 : L = analyzeStrand(F,a); #L -- 0.0416064 seconds elapsed o5 = 350 i6 : betti F_a, betti F 0 0 1 2 3 4 5 6 7 8 9 o6 = (total: 833, total: 1 36 187 491 793 833 573 250 63 7) 6: 350 0: 1 . . . . . . . . . 7: 468 1: . 36 160 342 436 350 174 49 6 . 8: 15 2: . . 27 148 351 468 379 186 51 6 3: . . . 1 6 15 20 15 6 1 o6 : Sequence i7 : factor product L 266 15 o7 = 2 3 o7 : Expression of class Product i8 : L3 = select(L,c->c%3==0); #L3 o9 = 14 i10 : carpetBettiTable(a,b,3) -- 0.00321821 seconds elapsed -- 0.00842124 seconds elapsed -- 0.0414264 seconds elapsed -- 0.0184893 seconds elapsed -- 0.00564361 seconds elapsed 0 1 2 3 4 5 6 7 8 9 o10 = 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 o10 : BettiTally

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

## Ways to use analyzeStrand :

• "analyzeStrand(ChainComplex,ZZ)"

## For the programmer

The object analyzeStrand is .