Classification of Six-Point Metrics

There are 339 combinatorial types of generic metrics on six points. They correspond to the 339 regular triangulations of the second hypersimplex ∆(6, 2), which also has 14 non-regular triangulations. 1 The Metric Fan We consider the cone of all metrics on the finite set {1, 2, . . . , n}: Cn = { d ∈ R(n2) : dij ≥ 0 and dij + djk ≥ dik for all 1 ≤ i, j, k ≤ n } . This is a closed convex pointed polyhedral cone. Its extreme rays have been studied in combinatorial optimization [4, 5]. Among the extreme rays are the splits. The splits are the metrics ∑ i∈A ∑ j 6∈A eij ∈ R( n 2) as A ranges over nonempty subsets of {1, 2, . . . , n}. There is an extensive body of knowledge (see [5, 9]) also on the facets of the subcone of Cn generated by the splits. Our object of study is a canonical subdivision of the metric cone Cn. It is called the metric fan and denoted MF n. A quick way to define the metric fan MF n is to say that it is the secondary fan of the second hypersimplex ∆(n, 2) = conv { ei + ej : 1 ≤ i < j ≤ n } ⊂ R. Every metric d defines a regular polyhedral subdivision ∆d of ∆(n, 2) as follows. The vertices of ∆(n, 2) are identified with the edges of the complete graph Kn, and subpolytopes of ∆(n, 2) correspond to arbitrary subgraphs of Kn. A subgraph G is a cell of ∆d if there exists an x ∈ R satisfying xi + xj = dij if {i, j} ∈ G and xi + xj > dij if {i, j} 6∈ G. the electronic journal of combinatorics 11 (2004), #R44 1 Two metrics d and d′ lie in the same cone of the metric fan MFn if they induce the same subdivision ∆d = ∆d′ of the second hypersimplex ∆(n, 2). We say that the metric d is generic if d lies in an open cone of MFn. This is equivalent to saying that ∆d is a regular triangulation of ∆(n, 2). These triangulation of ∆(n, 2) and the resulting metric fan MFn were studied by De Loera, Sturmfels and Thomas [3], who had been unaware of an earlier appearance of the same objects in phylogenetic combinatorics [1, 6]. In [6], Dress considered the polyhedron dual to the triangulation ∆d, Pd = { x ∈ R≥0 : xi + xj ≥ dij for 1 ≤ i < j ≤ n } , and he showed that its complex of bounded faces, denoted Td, is a natural object which generalizes the phylogenetic trees derived from the metric d. Both [3] and [6] contain the description of the metric fans MFn for n ≤ 5: • The octahedron ∆(4, 2) has three regular triangulations ∆d. They are equivalent up to symmetry. The corresponding tight span Td is a quadrangle with an edge attached to each of its four vertices. The three walls of the fan MF4 correspond to the trees on {1, 2, 3, 4}. • The fan MF5 has 102 maximal cones which come in three symmetry classes. The tight spans Td of these three metrics are depicted in [6, Figure A3], and the corresponding triangulations ∆d appear (in reverse order) in [3, page 414]. For instance, the thrackle triangulation of [3, §2] corresponds to the planar diagram in [6]. All three tight spans Td have five two-cells. (The type TX,D3 is slightly misdrawn in [6]: the two lower left quadrangles should form a flat pentagon). The aim of this article is to present the analogous classification for n = 6. The following result was obtained with the help of Rambau’s software TOPCOM [13] for enumerating triangulations of arbitrary convex polytopes. Theorem 1 There are 194, 160 generic metrics on six points. These correspond to the maximal cones in MF6 and to the regular triangulations of ∆(6, 2). They come in 339 symmetry classes. The hypersimplex ∆(6, 2) has also 3, 840 non-regular triangulations which come in 14 symmetry classes. This paper is organized as follows. In Section 2 we describe all 12 generic metrics whose tight span Td is two-dimensional, and in Section 3 we describe all 327 generic metrics whose tight span has a three-dimensional cell. Similarly, in Section 4, we describe the 14 non-regular triangulations of ∆(6, 2). In each case a suitable system of combinatorial invariants will be introduced. In Section 5 we study the geometry of the metric fan MF6. The rays of MF6 are precisely the prime metrics in [12]. We determine the maximal cones incident to each prime metric, and we discuss the corresponding minimal subdivisions of ∆(6, 2). In Section 6 we present a software tool for visualizing the tight span Td of any finite metric d. This tool was written written in POLYMAKE [10] with the help of the electronic journal of combinatorics 11 (2004), #R44 2 Michael Joswig and Julian Pfeifle. We also explain how its output differs from the output of SPLITSTREE [8]. A complete list of all six-point metrics has been made available at bio.math.berkeley.edu/SixPointMetrics For each of the 339+14 types in Theorem 1, the regular triangulation, Stanley-Reisner ideal, and numerical invariants are listed. The notation is consistent with that used in the paper. In addition, the webpage contains interactive pictures in JAVAVIEW [11] of the tight span of each metric. 2 The 12 Two-Dimensional Generic Metrics We identify each generic metric d with its tight span Td, where the exterior segments have been contracted so that every maximal cell has dimension ≥ 2. With this convention, generic four-point metrics are quadrangles and five-point metrics are glued from five polygons (cf. [6, Figure A3]). The generic six-point metrics, on the other hand, fall naturally into two groups. Lemma 2 Each generic metric on six points is either a three-dimensional cell complex with 26 vertices, 42 edges, 18 polygons and one 3-cell, or it is a two-dimensional cell complex with 25 vertices, 39 edges and 15 polygons. There are 327 three-dimensional metrics and 12 two-dimensional metrics. We first list the twelve types of two-dimensional metrics. In each case the tight span consists of 15 polygons which are either triangles, quadrangles or pentagons. Our first invariant is the vector B = (b3, b4, b5) where bi is the number of polygons with i sides. The next two invariants are the order of the symmetry group and the number of cubic generators in the Stanley-Reisner ideal of the triangulation ∆d. The last item is a representative metric d = (d12, d13, d14, d15, d16, d23, d24, d25, d26, d34, d35, d36, d45, d46, d56): Type 1: (1, 10, 4), 1, 2, (9, 9, 10, 13, 18, 18, 17, 6, 11, 17, 14, 9, 11, 8, 17) Type 2: (1, 10, 4), 1, 3, (8, 8, 8, 14, 15, 16, 14, 6, 9, 12, 12, 7, 8, 7, 13) Type 3: (1, 10, 4), 1, 5, (5, 6, 7, 8, 12, 11, 10, 5, 7, 11, 6, 6, 7, 5, 10) Type 4: (1, 10, 4), 2, 3, (7, 5, 7, 12, 12, 12, 12, 5, 7, 10, 9, 7, 7, 5, 10) Type 5: (1, 10, 4), 2, 4, (6, 7, 8, 10, 14, 13, 12, 6, 8, 13, 9, 7, 6, 6, 10) Type 6: (1, 10, 4), 2, 5, (7, 7, 7, 11, 14, 12, 12, 6, 7, 14, 10, 7, 6, 7, 11) Type 7: (1, 10, 4), 8, 6, (5, 5, 5, 8, 10, 10, 8, 5, 5, 8, 5, 5, 5, 5, 8) Type 8: (2, 8, 5), 1, 3, (5, 5, 7, 10, 11, 10, 10, 5, 8, 10, 7, 6, 5, 4, 7) Type 9: (2, 8, 5), 2, 4, (7, 7, 8, 10, 14, 14, 13, 5, 9, 13, 9, 7, 10, 6, 14) Type 10: (2, 8, 5), 2, 4, (5, 4, 5, 8, 9, 7, 8, 3, 6, 9, 6, 5, 5, 4, 7) Note that the exterior segments do appear in Figures 1–5 of this paper and in the diagrams on our webpage. They are drawn in green for extra clarity. the electronic journal of combinatorics 11 (2004), #R44 3 Type 11: (2, 8, 5), 2, 4, (4, 5, 5, 8, 9, 9, 7, 4, 7, 8, 5, 4, 5, 4, 7) Type 12: (3, 6, 6), 12, 3, (3, 3, 5, 6, 6, 6, 6, 3, 5, 6, 5, 3, 3, 3, 6) The three metrics of types 9, 10 and 11 cannot be distinguished by the given invariants. In Section 5 we explain how to distinguish these three types. The metric with the largest symmetry group is Type 12. Its symmetry group has order 12. This combinatorial type of this metric is given by the Stanley-Reisner ideal of the corresponding regular triangulation of ∆(6, 2): 〈x36x14, x25x34, x35x46, x16x45, x35x12, x26x35, x36x45, x15x36, x26x45, x12x46, x12x56, x25x36, x45x23, x24x13, x45x12, x34x12, x25x46, x23x46, x16x25, x13x46, x24x36, x35x14, x13x56, x26x14, x26x13, x15x46, x36x12, x45x13, x25x14, x25x13, x15x26x34, x23x56x14, x16x24x35 〉. The number of quadratic generators is 30, and this number is independent of the choice of generic metric. The number of cubic generators of this particular ideal is three (the last three generators), which is the third invariant listed under “Type 12”. These cubic generators correspond to “empty triangles” in the triangulation ∆d. For instance, the cubic x15x26x34 means that conv{e1 + e5, e2 + e6, e3 + e4} is not a triangle in ∆d but each of its three edges is an edge in ∆d. In the tight span Td this can be seen as follows: {geodesics between 1 and 5} ∩ {geodesics between 2 and 6} ∩ {geodesics between 3 and 4} = ∅, but any two of these sets of geodesics have a common intersection. This can be seen in the picture of the tight span of the type 12 metric in Figure 1. The twelve generic metrics listed above demonstrate the subtle nature of the notion of combinatorial dimension introduced in [6]. Namely, the combinatorial dimension of a generic metric d can be less than that of a generic split-decomposable metric [1]. This implies that the space of all n-point metrics of combinatorial dimension ≤ 2 is a polyhedral fan whose dimension exceeds the expected number 4n − 10 (cf. [7, Theorem 1.1 (d)]). For six-point metrics, this discrepancy can be understood by looking at the centroid ( 1 3 , 1 3 , 1 3 , 1 3 , 1 3 , 1 3 ) of the hypersimplex ∆(6, 2). There are 25 simplices in ∆(6, 2) which contain the centroid: the 15 triangles given by the perfect matchings of the graph K6 and the 10 five-dimensional simplices corresponding to two disjoint triangles in K6. In any given triangulation ∆d, the centroid can lie in either one or the other. In the former case, the tight span Td has a 3-dimensional cell dual to the perfect matching triangle in ∆d. The combinatorial possibilities of these 3-cells will be explored in Section 3. In the latter case, the tight span Td has a distinguished vertex dual to the two-disjoint-triangles simplex in ∆d. This vertex lies in nine polygons of Td which form a link of type K3,3. But there is no 3-cell in Td. The distinguished vertex is the one in the center in Figure 1. the electronic journal of combinatorics 11 (2004), #R44 4