arXiv research

We study some graphs associated to a surface, called k-multicurve graphs, which interpolate between the curve complex and the pants graph. Our main result is that, under certain conditions, simplicial embeddings between multicurve graphs are induced by $π_1$-injective embeddings of the corresponding surfaces. We also p…

A non-separating multicurve of a surface S of genus g with m punctures is a multicurve c so that S-c is connected. For k>0 define the graph of non-separting k-multicurves to be the graph whose vertices are non-separating multicurves with k components and where two such multicurves are connected by an edge if they can b…

Graphs of multicurves are hyperbolic, relatively hyperbolic, or thick.

Extends Teichmüller space quasi-isometry to $k$-multicurve graphs.

Graphs of multicurves are hierarchically hyperbolic spaces.

Two graphs on infinite-type surfaces have finite diameter and related automorphism groups.

Maps between Hatcher-Thurston graphs show genus restrictions and multicurve properties.

We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense. We show that the graphs we construct are generally connected, infinite diameter a…

Algorithm calculates multicurve intersections using Dynnikov coordinates.

Research examines the distribution of curve components in random multicurves.

Simple model prices swaptions in multicurve interest rates.

Khovanov multicurves are restricted to linear components.

Formula for Heegaard Floer multicurves of double tangles from knot complements.

Multicurve stabilizers' extensions are hierarchically hyperbolic.

This paper explores how pairs of multicurves can be realized as cylinders on translation surfaces.

This note is about the geometry of the pants graph P(S), a natural simplicial graph associated to a finite type topological surface S where vertices represents pants decompositions. The main result in this note ascserts that for a multicurve Q whose complement is a number of subsurfaces of complexity at most 1. We prov…

Study on lengths of random multicurves on hyperbolic surfaces.

New graphs show hierarchical hyperbolic properties, extending previous work.

Study on length distribution of random multicurves on large genus surfaces converging to Poisson-Dirichlet distribution.

Study of graphs interpolating curve and pants graphs, providing formulae and geometry classifications.

Proves cosmetic surgery conjecture for strongly invertible knots.

Polynomial bound on tightening curves on surfaces without increasing crossings.

Study of random multicurves and square-tiled surfaces on large genus surfaces.

Characterizes curves with short representatives on hyperbolic surfaces.

Given a set equipped with a transitive action of a group, we define the notion of an almost invariant coloring of the set. We consider the mapping class group orbit of a multicurve on a compact surface, and prove that in the case of genus at least two, no such almost invariant coloring exists. Conversely, in the case o…

The study examines translation lengths of pseudo-Anosov maps on curve graphs.

Study shows surfaces' moduli spaces have special geometric properties.

Study crosscap transpositions on non-orientable surfaces, providing explicit formulae.

We consider the representation space of a compact surface, that is the space of morphisms from the fundamental group to SU(2) up to conjugation. We show that the trace functions associated to multicurves on the surface are linearly independent as functions on the representation space. The proof relies on the Fourier de…

Various curve complexes with vertices representing multicurves on a surface $S$ have been defined, for example [3], [4] and [8]. The homology curve complex $\mathcal{HC}(S,α)$ defined in [7] is one such complex, with vertices corresponding to multicurves in a nontrivial integral homology class $α$. Given two multicurve…

A \textit{multicurve} $\C$ on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist $t_{\C}$ about $\C$ is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the exi…

New graph invariant shows complexity of surface graphs.

Characterizes pseudo-Anosov mapping classes on general marked surfaces.

Let $X$ be a closed hyperbolic surface and $λ, η$ be weighted geodesic multicurves which are short on X. We show that the iterated grafting along $λ$ and $η$ is close in the Teichmueller metric to grafting along a single multicurve which can be given explicitly in terms of $λ$ and $η$. Using this result, we study the h…

The musical notes from a hyperbolic marimba can identify the shape of hyperbolic surfaces.

Study volumes of hyperbolic 3-manifolds formed by curves on surfaces.

The study of which mapping class group elements can be realized as affine automorphisms of dilation surfaces.

On a surface with a Finsler metric, we investigate the asymptotic growth of the number of closed geodesics of length less than $L$ which minimize length among all geodesic multicurves in the same homology class. An important class of surfaces which are of interest to us are hyperbolic surfaces.

New quandles classify surfaces, with infinite cohomology.

A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston's boundary of Teichmüller space.

For a fixed marked surface $S$, we show that the problem of deciding whether or not a mapping class is reducible lies in $\textbf{NP}$. As usual this immediately gives an exponential time algorithm to decide whether or not a mapping class is reducible. To do this we use an (ideal) triangulation to obtain a coordinate s…

Formulae for Masur-Veech volumes derived from intersection numbers of curves.

Khovanov homology invariant under Conway mutation.

We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A|<1, and only densel…

Geometric interpretation of tangle invariants using immersed curves.

Study proves hyperbolic structures for link complements in Seifert fibered spaces.

We calculate volumes of quadratic differentials using topological recursion.

Random simple closed curves map Teichmüller space to geodesic currents.