polymake: a Framework for Analyzing Convex Polytopes

Gawrilow, Ewgenij; Joswig, Michael

doi:10.1007/978-3-0348-8438-9_2

Cited by 516 publications

(591 citation statements)

References 9 publications

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“…As a special case, we show that Trop + Gr 2,n is a fan which appeared in the work of Stanley and Pitman (see [19]), which parameterizes certain binary trees, and which is combinatorially equivalent to the (type A n ) associahedron. We also show that Trop + Gr 3,6 and Trop + Gr 3,7 are fans which are closely related to the fans of the types D 4 and E 6 associahedra, which were first introduced in [5]. These results are strikingly reminiscent of the results of Fomin and Zelevinsky [3], and Scott [17], who showed that the Grassmannian has a natural cluster algebra structure which is of type A n for Gr 2,n , type D 4 for Gr 3,6 , and type E 6 for Gr 3,7 .…”

Section: Introductionsupporting

confidence: 66%

The Tropical Totally Positive Grassmannian

2005

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Abstract. Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). The theory of total positivity is a natural generalization of the study of matrices with all minors positive. In this paper we introduce the totally positive part of the tropicalization of an arbitrary affine variety, an object which has the structure of a polyhedral fan. We then investigate the case of the Grassmannian, denoting the resulting fan Trop + Gr k,n . We show that Trop + Gr 2,n is the Stanley-Pitman fan, which is combinatorially the fan dual to the (type A n−3 ) associahedron, and that Trop + Gr 3,6 and Trop + Gr 3,7 are closely related to the fans dual to the types D 4 and E 6 associahedra. These results are strikingly reminiscent of the results of Fomin and Zelevinsky, and Scott, who showed that the Grassmannian has a natural cluster algebra structure which is of types A n−3 , D 4 , and E 6 for Gr 2,n , Gr 3,6 , and Gr 3,7 . We suggest a general conjecture about the positive part of the tropicalization of a cluster algebra.

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Section: Introductionsupporting

confidence: 66%

The Tropical Totally Positive Grassmannian

2005

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“…The computations in this paper were done with a combination of Gfan (Jensen 2009), Polymake (Gawrilow and Joswig 2000), and custom ocaml (Chailloux et al 2000) code using GSL (Galassi et al 2009), the GNU scientific library. For the interested reader, source code is available at http://github.com/matsen/roguebme.…”

Section: Overview Of the Papermentioning

confidence: 99%

Polyhedral Geometry of Phylogenetic Rogue Taxa

Cueto

Matsen

2010

Bull Math Biol

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It is well known among phylogeneticists that adding an extra taxon (e.g. species) to a data set can alter the structure of the optimal phylogenetic tree in surprising ways. However, little is known about this "rogue taxon" effect. In this paper we characterize the behavior of balanced minimum evolution (BME) phylogenetics on data sets of this type using tools from polyhedral geometry. First we show that for any distance matrix there exist distances to a "rogue taxon" such that the BME-optimal tree for the data set with the new taxon does not contain any nontrivial splits (bipartitions) of the optimal tree for the original data. Second, we prove a theorem which restricts the topology of BME-optimal trees for data sets of this type, thus showing that a rogue taxon cannot have an arbitrary effect on the optimal tree. Third, we computationally construct polyhedral cones that give complete answers for BME rogue taxon behavior when our original data fits a tree on four, five, and six taxa. We use these cones to derive sufficient conditions for rogue taxon behavior for four taxa, and to understand the frequency of the rogue taxon effect via simulation.

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“…In a second step we do a pulling triangulation along a vertex order given by a (globally fixed) linear functional. The analysis of the cells was done using the software package polymake [GJ05].…”

Section: Quadratic Gröbner Basesmentioning

confidence: 99%

Quadratic Gröbner bases for smooth 3×3 transportation polytopes

Haase

Paffenholz

2009

J Algebr Comb

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Abstract. The toric ideals of 3 × 3 transportation polytopes T rc are quadratically generated. The only exception is the Birkhoff polytope B 3 .If T rc is not a multiple of B 3 , these ideals even have squarefree quadratic initial ideals. This class contains all smooth 3 × 3 transportation polytopes.1. IntroductionIf X P is smooth (the normal fan of P is unimodular), then L P is very ample, and provides an embedding X P ֒→ P r−1 , where r = #(P ∩ d ). So we can think of X P as canonically sitting in projective space. The following question [Stu97, Conjecture 2.9] about the defining equations of X P ⊂ P r−1 has been around for quite a while, but its origins are hard to track (cf. [BCF + 05]).Question. Let P be a lattice polytope whose corresponding projective toric variety is smooth. Is the defining ideal I P generated by quadratics?There are two variations of this question (which are of strictly increasing strength).• Is the homogeneous coordinate ring [x 1 , . . . , x r ]/I P Koszul?• Does I P have a quadratic Gröbner basis? The last version has a combinatorial interpretation. It asks for the existence of very special, "quadratic" triangulations of P , see §1.3 below.1.2. Results. Simple transportation polytopes provide a large family of smooth polytopes. Yet, the 3 × 3 Birkhoff polytope B 3 is a nonsimple transportation polytope whose ideal is not generated by quadratic polynomials. In this note, we show that in the 3 × 3 case, this is the only example. In Section 2, we show that B 3 is the only (3 × 3)-transportation polytope whose ideal is not quadratically generated.

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polymake: a Framework for Analyzing Convex Polytopes

Cited by 516 publications

References 9 publications

The Tropical Totally Positive Grassmannian

The Tropical Totally Positive Grassmannian

Polyhedral Geometry of Phylogenetic Rogue Taxa

Quadratic Gröbner bases for smooth 3×3 transportation polytopes

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