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K3Carpets :: homotopyRanks

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

Synopsis

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 :

For the programmer

The object homotopyRanks is a method function with options.