Supplementary Materials for "Identifiability of 3-class Jukes-Cantor Mixtures"

 This webpage contains supplementary materials for the paper Identifiability of 3-class Jukes Cantor mixtures by Colby Long and Seth Sullivant. The proof of Theorem 1.1 that establishes the identifiability result involves computing phylogenetic invariants for various 3-class mixtures. Files allowing one to verify these computations are available on this page for download; a rigorous justification for why these computations suffice to prove Theorem 1.1 can be found in the paper. The supplementary files were all generated using the computer algebra software Maple 14 and can be executed by opening and running them in Maple. The file LinearInvariants_5Leaf.mw constructs all 3-class mixtures on 5-leaf trees and calculates the linear invariant space associated to each triplet in Fourier coordinates. Each triplet pair is then compared and the result of the computation is the set AllFiveLeafPairs. This list contains all pairs of 5-leaf triplets (up to the action of S_5) which share the same linear invariant space. The file LinearInvariants_6Leaf.mw generates all pairs of 3-class mixtures on 6-leaf trees that can be built up from the pairs of triplets in AllFiveLeafPairs. As outlined in the paper, it is only necessary to compare the linear invariant space of 6-leaf triplets constructed in this manner. The result is the list AllSixLeafPairs containing all pairs of 6-leaf triplets (up to the action of S_6) which share the same linear invariant space. Section 6.2 outlines an elimination computation for finding separating higher degree invariants for the mixture models contained in AllSixLeafPairs. For each of the 85 pairs of 6-leaf triplets in AllSixLeafPairs, an elimination in Macaulay2 was used to find an invariant separating the triplets. The Maple file Higher_Degree_Invariants.mw lists one such separating invariant separating each triplet pair. The provided code quickly generates the parameterization of the coordinate functions to show that the proposed invariant evaluates to zero for exactly one of the triplets. Both LinearInvariants_5Leaf.mw and Higher_Degree_Invariants.mw run in few minutes on a laptop. LinearInvariants_6Leaf.mw takes about 2 days to finish the computation on a laptop.