Related papers: Loops in surfaces and star-fillings
In the mid eighties Goldman proved an embedded curve could be isotoped to not intersect a closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the…
We relate the author's Lie cobracket in the module additively generated by loops on a surface with the Connes-Kreimer Lie bracket in the module additively generated by trees. To this end we introduce a pre-Lie coalgebra and a (commutative)…
In the context of (2+1)--dimensional gravity, we use holonomies of constant connections which generate a $q$--deformed representation of the fundamental group to derive signed area phases which relate the quantum matrices assigned to…
In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points…
In this paper we show that, after completion in the I-adic topology, the Goldman bracket on the space spanned by homotopy classes of loops on a smooth, complex algebraic curve is a morphism of mixed Hodge structure. We prove similar…
The mapping class group of a surface $\S$ acts on the set of closed geodesics on $\S$. This action preserves self-intersection number. In this paper, we count the orbits of curves with at most $K$ self-intersections, for each $K \geq 1$.…
In two seminal papers Kontsevich used a construction called_graph homology_ as a bridge between certain infinite dimensional Lie algebras and various topological objects, including moduli spaces of curves, the group of outer automorphisms…
We extend the existing idea of "cellular poset", introduce a collection of "cellular methods" for the computation of homology of intersection lattice of a complicated subspace arrangement, and for the computation of multiplicative structure…
By introducing an invariant of loops on a compact oriented surface with one boundary component, we give an explicit formula for the action of Dehn twists on the completed group ring of the fundamental group of the surface. This invariant…
W. Goldman and V. Turaev defined a Lie bialgebra structure on the $\mathbb Z$-module generated by free homotopy classes of loops of an oriented surface (i.e. the conjugacy classes of its fundamental group). We develop a generalization of…
We provide some explicit algebraic criteria in terms of the Goldman bracket to decide whether two free homotopy classes of loops on an oriented surface admit disjoint representatives. We extend Kabiraj's method using the hyperbolic geometry…
In this note we develop a tool box of non-Euclidean plane geometry methods that yield a constructive way to define in terms of closed geodesics the Goldman bracket on deformation classes of closed, directed curves. We use this construction…
We introduce an operation that measures the self intersections of paths on a surface. As applications, we give a criterion of the realizability of a generalized Dehn twist, and derive a geometric constraint on the image of the Johnson…
In this paper, we develop a new and efficient approach to the computation of envelope surfaces. We interpret one-parameter systems of surfaces as curves in the homogeneous spaces of suitable Lie groups. Using the formalism of Lie groups and…
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
The free $\mathbb{Z}$-module generated from the set of non-trivial homotopy classes of closed curves on an oriented surface has the structure of Lie bialgebra by two operations, the Goldman bracket and Turaev cobracket. M. Chas gave a…
The Turaev cobracket, a loop operation introduced by V. Turaev, which measures self-intersection of a loop on a surface, is a modification of a path operation introduced earlier by Turaev himself, as well as a counterpart of the Goldman…
Given a compact orientable surface $\Sigma$, let $\Cal S(\Sigma)$ be the set of isotopy classes of essential simple loops on $\Sigma$. We determine a complete set of relations for a function from $\Cal S(\Sigma)$ to $\bold Z$ to be a…
Based on Nielsen fixed point theory and Gr\"{o}bner-Shirshov basis, we obtain a simple method to compute geometric intersection numbers and self-intersection geometric numbers of loops on surfaces.
Goldman and Turaev found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in…