Splicing matroids

We introduce and study a natural variant of matroid amalgams. For matroids M ( A ) and N ( B ) with M . ( A ∩ B ) = N | ( A ∩ B ) , we define a splice of M and N to be a matroid L on A ∪ B with L | A = M and L . B = N . We show that splices exist for each such pair of matroids M and N ; furthermore, there is a freest splice of M and N , which we call the free splice. We characterize when a matroid L ( A ∪ B ) is the free splice of L | A and L . B . We study minors of free splices and the interaction between free splice and several other matroid operations. Although free splice is not an associative operation, we prove a weakened counterpart of associativity that holds in general and we characterize the triples for which associativity holds. We also study free splice as it relates to various classes of matroids.