# homotopyRanks -- compute the ranks of the quadratic homotopies on a carpet

## Synopsis

• Usage:
L = homotopyRanks(g,cliff)
• Inputs:
• g, an integer, genus of the carpet
• cliff, an integer, Clifford index of the carpet
• Optional inputs:
• Characteristic => an integer, default value 32003, the characteristic of the ground field
• Outputs:
• L, a net, netList showing the ranks

## Description

Prints the Betti table of the canonical carpet and a list of pairs, with first element a quadric and second element the sequence of ranks of the homotopies for that quadric on F_i, for i = 1..g-2.

 i1 : homotopyRanks(7,3) 0 1 2 3 4 5 total: 1 10 16 16 10 1 0: 1 . . . . . 1: . 10 16 . . . 2: . . . 16 10 . 3: . . . . . 1 +-------------------------+---------------+ o1 = || x_1^2-x_0x_2 | |{1, 0, 8, 0, 1}| +-------------------------+---------------+ || x_1x_2-x_0x_3 | |{1, 0, 8, 0, 1}| +-------------------------+---------------+ || x_2^2-x_1x_3 | |{1, 0, 8, 0, 1}| +-------------------------+---------------+ || x_2y_0-2x_1y_1+x_0y_2 ||{1, 0, 8, 0, 1}| +-------------------------+---------------+ || x_3y_0-2x_2y_1+x_1y_2 ||{1, 0, 8, 0, 1}| +-------------------------+---------------+ || x_2y_1-2x_1y_2+x_0y_3 ||{1, 0, 8, 0, 1}| +-------------------------+---------------+ || x_3y_1-2x_2y_2+x_1y_3 ||{1, 0, 8, 0, 1}| +-------------------------+---------------+ || y_1^2-y_0y_2 | |{1, 0, 8, 0, 1}| +-------------------------+---------------+ || y_1y_2-y_0y_3 | |{1, 0, 8, 0, 1}| +-------------------------+---------------+ || y_2^2-y_1y_3 | |{1, 0, 8, 0, 1}| +-------------------------+---------------+ i2 : homotopyRanks(7,3, Characteristic => 2) 0 1 2 3 4 5 total: 1 10 19 19 10 1 0: 1 . . . . . 1: . 10 16 3 . . 2: . . 3 16 10 . 3: . . . . . 1 +-----------------+---------------+ o2 = || x_1^2+x_0x_2 | |{1, 0, 0, 2, 1}| +-----------------+---------------+ || x_1x_2+x_0x_3 ||{1, 0, 0, 0, 1}| +-----------------+---------------+ || x_2^2+x_1x_3 | |{1, 0, 0, 2, 1}| +-----------------+---------------+ || x_2y_0+x_0y_2 ||{1, 0, 0, 0, 1}| +-----------------+---------------+ || x_3y_0+x_1y_2 ||{1, 0, 0, 0, 1}| +-----------------+---------------+ || x_2y_1+x_0y_3 ||{1, 0, 0, 0, 1}| +-----------------+---------------+ || x_3y_1+x_1y_3 ||{1, 0, 0, 0, 1}| +-----------------+---------------+ || y_1^2+y_0y_2 | |{1, 0, 0, 2, 1}| +-----------------+---------------+ || y_1y_2+y_0y_3 ||{1, 0, 0, 0, 1}| +-----------------+---------------+ || y_2^2+y_1y_3 | |{1, 0, 0, 2, 1}| +-----------------+---------------+

## Ways to use homotopyRanks :

• "homotopyRanks(ZZ,ZZ)"

## For the programmer

The object homotopyRanks is .