Deformations of the braid arrangement and trees (original) (raw)

The braid monodromy of plane algebraic curves and hyperplane arrangements

Commentarii Mathematici Helvetici, 1997

To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin's braid group B n . Using Hansen's polynomial covering space theory, we give a new interpretation of this construction. Next, we provide an explicit description of the braid monodromy of an arrangement of complex affine hyperplanes, by means of an associated "braided wiring diagram." The ensuing presentation of the fundamental group of the complement is shown to be Tietze-I equivalent to the Randell-Arvola presentation. Work of Libgober then implies that the complement of a line arrangement is homotopy equivalent to the 2-complex modeled on either of these presentations. Finally, we prove that the braid monodromy of a line arrangement determines the intersection lattice. Examples of Falk then show that the braid monodromy carries more information than the group of the complement, thereby answering a question of Libgober.

On an open problem of Green and Losonczy: exact enumeration of freely braided permutations

Discrete Mathematics & Theoretical Computer Science, 2004

International audience Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par

Bijections between affine hyperplane arrangements and valued graphs

European Journal of Combinatorics, 2015

We show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-brokencircuit sets of the corresponding integral gain graphs and some kinds of labelled binary trees. This leads to new bijective proofs for the Shi, Catalan, and similar hyperplane arrangements.

Braid monodromy of complex line arrangements

Kodai Mathematical Journal, 1999

Let V be the complex vector space C 7 , s/ an arrangement in V, i.e. a finite family of hyperplanes in V In , Moishezon associated to any algebraic plane curve <# of degree n a braid monodromy homomorphism θ F s -> B(n), where F s is a free group, B(ή) is the Artm braid group. In this paper, we will determine the braid monodromy for the case when # is an arrangement stf of complex lines in C 2 , using the notion of labyrinth of an arrangement. As a corollary we get the braid monodromy presentation for the fundamental group of the complement to the arrangement.

The braid and the Shi arrangements and the Pak–Stanley labelling

European Journal of Combinatorics, 2015

In this article we study a construction, due to Pak and Stanley, with which every region R of the Shi arrangement is (bijectively) labelled with a parking function λ(R). In particular, we construct an algorithm that returns R out of λ(R). This is done by relating λ to another bijection, that labels every region S of the braid arrangement with r(S), the unique central parking function f such that λ −1 (f) ⊆ S. We also prove that λ maps the bounded regions of the Shi arrangement bijectively onto the prime parking functions. Finally, we introduce a variant (that we call "s-parking") of the parking algorithm that is in the very origin of the term "parking function". Sparking may be efficiently used in the context of our new algorithm, but we show that in some (well defined) cases it may even replace it.

A generalized Desargues configuration and the pure braid group

Discrete Mathematics, 1996

In this paper, a configuration with n = (g) points in the plane is described. This configuration, as a matroid, is a Desargues configuration if d = 5, and the union of (~) such configurations if d > 5. As an oriented matroid, it is a rank 3 truncation of the directed complete graph on d vertices. From this fact, it follows from a version of the Lefschetz Zariski theorem implied by results of Salvetti that the fundamental group ~ of the complexification of its line arrangement is Artin's pure (or coloured) braid group on d strands. In this paper we obtain, by using techniques introduced by Salvetti, a new algorithm for finding a presentation of ~ based on this particular configuration.

Decoration invariants for horseshoe braids

Discrete and Continuous Dynamical Systems, 2010

The Decoration Conjecture describes the structure of the set of braid types of Smale's horseshoe map ordered by forcing, providing information about the order in which periodic orbits can appear when a horseshoe is created. A proof of this conjecture is given for the class of so-called lone decorations, and it is explained how to calculate associated braid conjugacy invariants which provide additional information about forcing for horseshoe braids.

Building planar polygon spaces from the projective braid arrangement

Cornell University - arXiv, 2022

A. The moduli space of planar polygons with generic side lengths is a smooth, closed manifold. It is known that these manifolds contain the real points of the moduli space of distinct points on the projective line as an open dense subset. Hence, such a polygon space is a compactification of this real moduli space. Kapranov showed that the real points of the Deligne-Mumford-Knudson compactification can be obtained from the projective Coxeter complex of type A (equivalently, the projective braid arrangement) by iteratively blowing up along the minimal building set. In this paper we show that these planar polygon spaces can also be obtained from the projective Coxeter complex of type A by performing an iterative cellular surgery along a sub-collection of the minimal building set. Interestingly, this sub-collection is determined by the combinatorial data associated with the length vector called the genetic code.