A combinatorial approach to determining phylogenetic invariants for the general model

We present a combinatorial approach for calculating the phylogenetic invariants for an evolutionary tree T with n taxa when one is given molecular sequences consisting of k bases. Such invariants are potentially useful for reconstructing the evolutionary tree of n taxa from sequence data. We show that the homogeneous invariants for the general, stochastic Markov model of sequence evolution are the same as those for the general model. We show that homogeneous invariants of degree d can only arise when d ≥ k+1. We also calculate the invariants of low-degree when k = 2, k = 4 for the general model. When k = 2, we provide a theoretical explanation for the cubic invariants found by [5] for the four-leaf tree. We can also show that these cubic invariants generate the quartic and quintic invariants when k = 2 for the four-leaf tree for the general model. When k = 4, we determine all the quintic invariants when n = 3 and give a constructive construction of 19, 079, 424 quintic invariants when n = 4 for the general 12-parameter model. Résume