Operads and Phylogenetic Trees

We construct an operad $\mathrm{Phyl}$ whose operations are the edge-labelled trees used in phylogenetics. This operad is the coproduct of $\mathrm{Com}$, the operad for commutative semigroups, and $[0,\infty)$, the operad with unary operations corresponding to nonnegative real numbers, where composition is addition. We show that there is a homeomorphism between the space of $n$-ary operations of $\mathrm{Phyl}$ and $\mathcal{T}_n\times [0,\infty)^{n+1}$, where $\mathcal{T}_n$ is the space of metric $n$-trees introduced by Billera, Holmes and Vogtmann. Furthermore, we show that the Markov models used to reconstruct phylogenetic trees from genome data give coalgebras of $\mathrm{Phyl}$. These always extend to coalgebras of the larger operad $\mathrm{Com} + [0,\infty]$, since Markov processes on finite sets converge to an equilibrium as time approaches infinity. We show that for any operad $O$, its coproduct with $[0,\infty]$ contains the operad $W(O)$ constucted by Boardman and Vogt. To prove these results, we explicitly describe the coproduct of operads in terms of labelled trees.Comment: 48 pages, 3 figure

On surgery along Brunnian links in 3-manifolds

We consider surgery moves along (n+1)-component Brunnian links in compact connected oriented 3-manifolds, where the framing of the each component is 1/k for k in Z. We show that no finite type invariant of degree < 2n-2 can detect such a surgery move. The case of two link-homotopic Brunnian links is also considered. We relate finite type invariants of integral homology spheres obtained by such operations to Goussarov-Vassiliev invariants of Brunnian links.Comment: This is the version published by Algebraic & Geometric Topology on 13 December 200

Automorphism Groups of Geometrically Represented Graphs

We describe a technique to determine the automorphism group of a geometrically represented graph, by understanding the structure of the induced action on all geometric representations. Using this, we characterize automorphism groups of interval, permutation and circle graphs. We combine techniques from group theory (products, homomorphisms, actions) with data structures from computer science (PQ-trees, split trees, modular trees) that encode all geometric representations. We prove that interval graphs have the same automorphism groups as trees, and for a given interval graph, we construct a tree with the same automorphism group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982]. For permutation and circle graphs, we give an inductive characterization by semidirect and wreath products. We also prove that every abstract group can be realized by the automorphism group of a comparability graph/poset of the dimension at most four

Tree homology and a conjecture of Levine

In his study of the group of homology cylinders, J. Levine made the conjecture that a certain homomorphism eta': T -> D' is an isomorphism. Here T is an abelian group on labeled oriented trees, and D' is the kernel of a bracketing map on a quasi-Lie algebra. Both T and D' have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory, and the homology of the group of automorphisms of the free group. In this paper, we confirm Levine's conjecture. This is a central step in classifying the structure of links up to grope and Whitney tower concordance, as explained in other papers of this series. We also confirm and improve upon Levine's conjectured relation between two filtrations of the group of homology cylinders

The Algebra of Binary Search Trees

We introduce a monoid structure on the set of binary search trees, by a process very similar to the construction of the plactic monoid, the Robinson-Schensted insertion being replaced by the binary search tree insertion. This leads to a new construction of the algebra of Planar Binary Trees of Loday-Ronco, defining it in the same way as Non-Commutative Symmetric Functions and Free Symmetric Functions. We briefly explain how the main known properties of the Loday-Ronco algebra can be described and proved with this combinatorial point of view, and then discuss it from a representation theoretical point of view, which in turns leads to new combinatorial properties of binary trees.Comment: 49 page

Affine actions on non-archimedean trees

We initiate the study of affine actions of groups on $\Lambda$-trees for a general ordered abelian group $\Lambda$; these are actions by dilations rather than isometries. This gives a common generalisation of isometric action on a $\Lambda$-tree, and affine action on an $\R$-tree as studied by I. Liousse. The duality between based length functions and actions on $\Lambda$-trees is generalised to this setting. We are led to consider a new class of groups: those that admit a free affine action on a $\Lambda$-tree for some $\Lambda$. Examples of such groups are presented, including soluble Baumslag-Solitar groups and the discrete Heisenberg group.Comment: 27 pages. Section 1.4 expanded, typos corrected from previous versio

The Weight Function in the Subtree Kernel is Decisive

Tree data are ubiquitous because they model a large variety of situations, e.g., the architecture of plants, the secondary structure of RNA, or the hierarchy of XML files. Nevertheless, the analysis of these non-Euclidean data is difficult per se. In this paper, we focus on the subtree kernel that is a convolution kernel for tree data introduced by Vishwanathan and Smola in the early 2000's. More precisely, we investigate the influence of the weight function from a theoretical perspective and in real data applications. We establish on a 2-classes stochastic model that the performance of the subtree kernel is improved when the weight of leaves vanishes, which motivates the definition of a new weight function, learned from the data and not fixed by the user as usually done. To this end, we define a unified framework for computing the subtree kernel from ordered or unordered trees, that is particularly suitable for tuning parameters. We show through eight real data classification problems the great efficiency of our approach, in particular for small datasets, which also states the high importance of the weight function. Finally, a visualization tool of the significant features is derived.Comment: 36 page