spike -- spike matroid

The family of spikes appears in Oxley, p. 661-662. For a given r, every r-spike has 2r+1 elements and rank r. The ground set consists of a tip 0, and r legs {0, 1, 2}, {0, 3, 4}, ..., {0, 2*r-1, 2*r} (each of which is a circuit).

Deleting the tip 0 gives a matroid of rank r on 2r elements called a tipless r-spike.

The optional input L should be a list of additional circuits, subject to certain additional conditions to be a spike. If no additional circuits are provided, then the spike is called free.

Out of all possible r-spikes, there is a a unique one which is binary (i.e. representable over the field of 2 elements): this is returned by the function binarySpike.

i1 : M = binarySpike 5

o1 = a "matroid" of rank 5 on 11 elements

o1 : Matroid

i2 : getRepresentation M

o2 = | 1 0 0 0 0 0 1 1 1 1 1 |
     | 0 1 0 0 0 1 0 1 1 1 1 |
     | 0 0 1 0 0 1 1 0 1 1 1 |
     | 0 0 0 1 0 1 1 1 0 1 1 |
     | 0 0 0 0 1 1 1 1 1 0 1 |

             ZZ 5      ZZ 11
o2 : Matrix (--)  <-- (--)
              2         2

i3 : N = M \ set{0}

o3 = a "matroid" of rank 5 on 10 elements

o3 : Matroid

i4 : areIsomorphic(N, dual N)

o4 = true

i5 : N1 = (spike 5) \ set{0}

o5 = a "matroid" of rank 5 on 10 elements

o5 : Matroid

i6 : N1 == dual N1

o6 = true