H-Representation of the Kimura-3 Polytope for the $m$-Claw Tree

Abstract

Given a group-based Markov model on a tree, one can compute the vertex representation of a polytope describing a toric variety associated to the algebraic statistical model. In the case of Z 2 or Z 2 × Z 2 , these polytopes have applications in the field of phylogenetics. We provide a half-space representation for the m-claw tree where G = Z 2 × Z 2 , which corresponds to the Kimura-3 model of evolution.

Gorenstein property for phylogenetic trivalent trees

“…The vertex description for polytopes representing group-based models is already known. On the contrary, the facet description is known only in few cases, namely in case of G = Z 2 and G = Z 2 × Z 2 , see [20]. In general, it is a difficult problem to obtain the facet description from the vertex description.…”

“…Thus, in all cases we can denote by ω G,m the only possible candidate. We have the inequalities for P G,m from [20] and Theorem 2.2. Now we compute the lattice distances from ω G,m to the facets of P G,m and conclude our desired result.…”

Normality of the Kimura 3-parameter model

“…Equivalent characterization of P n is given in [MRV17]. The polytope is defined by the following inequalities:…”

“…However, obtaining facet description from the vertex one is hard in the general case, and for phylogenetic models in particular. For the 3-Kimura model such a description was provided in [MRV17]. First two results of the above allow us to translate the question about projective normality of the variety associated to the 3-Kimura model into a purely combinatorial statement about normality of a family of polytopes.…”

Finite phylogenetic complexity ofZpand invariants forZ3Michałek1 2017European Journal of Combinatorics2010View full textAdd to dashboardCite

“…We would like also to mention that the varieties X(T, G) share many other very interesting algebraic and combinatorial properties related to their Hilbert polynomial, normality and deformations [BBKM13,BW07,Kub12,MRV14].…”