arXiv research

Study on acceptable bundles on a punctured disk.

Using the techniques developed in \cite{SunSun}, we give estimations of the Bergman kernel of the punctured disk with the standard complete Poincaré metric. As an application, we improve the result of \cite{AMM} on the Bergman kernels of punctured Riemann surfaces near singularities.

For a smooth immersion $f$ from the punctured disk $D\backslash\{0\}$ into $\mathbb{R}^n$ extendable continuously at the puncture, if its mean curvature is square integrable and the measure of $f(D)\cap B_{r_k}=o(r_k)$ for a sequence $r_k\to 0$, we show that the Riemannian surface $(D_r\backslash\{0\},g)$ where $g$ is …

New Khovanov homology for links with multiple punctures.

Hyperbolic links in handlebodies can be composed, unlike in 3-sphere.

Algorithm detects free products in disk mapping class groups.

Study on the minimum length of curves on once-punctured hyperbolic surfaces.

We generalize Dynnikov coordinate system previosly defined on the standard punctured disk to an orientable surface of genus-1 with n punctures and one boundary component.

We give a recipe to compute the geometric intersection number of an integral lamination with a particular type of integral lamination on an n-times punctured disk. This provides a way to find the geometric intersection number of two arbitrary integral laminations when combined with an algorithm of Dynnikov and Wiest.

Let $D_n$ be the $n$-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most $\binom{n+1}{3}$. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs inter…

It was recently shown that the Carathéodory and Teichmüller metrics on the Teichmüller space of a closed surface do not coincide. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichmüller disk generated by a differential with no odd-order zeros. Our aim is to classify Teichmüller …

We show that the group cohomology of the diffeomorphisms of the disk with $n$ punctures has the cohomology of the braid group of $n$ strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding $\mathrm{Br}_{2g+2}\hookrightarrow \…

Study shows certain knots cannot bound disks in specific 4-manifolds.

Augmented alternating links are links obtained by adding trivial components that bound twice-punctured disks to non-split reduced non-2-braid prime alternating projections. These links are known to be hyperbolic. Here, we extend to show that generalized augmented alternating links, which allow for new trivial component…

We study the maximal entropy per unit generator of push-point mapping classes on the punctured disk. Our work is motivated by fluid mixing by rods in a planar domain. If a single rod moves among N-fixed obstacles, the resulting fluid diffeomorphism is in the push-point mapping class associated with the loop in π_1(D^2 …

It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is given, there is an algorithm to determine if the slope bounds an embedded punctured-d…

New homomorphism proven using immersed curves on disks.

A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehor…

We show that a smooth radially symmetric solution $u$ to the graphic Willmore surface equation is either a constant or the defining function of a half sphere in ${\mathbb R}^3$. In particular, radially symmetric entire Willmore graphs in ${\mathbb R}^3$ must be flat. When $u$ is a smooth radial solution over a puncture…

A new method converts knot Floer homology to immersed curves.

In this paper, we show that the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.

We give a presentation for the baseleaf preserving mapping class group $Mod(§)$ of the punctured solenoid $§$. The generators for our presentation were introduced previously, and several relations among them were derived. In addition, we show that $Mod(§)$ has no non-trivial central elements. Our main tool is a new com…

We classify right-veering homeomorphisms of the once-punctured torus using the Burau representation of the 3-strand braid group. We show that reducible and periodic mapping classes in B_3 can be identified as right-veering by consideration of the reduced version of the Burau representation. Given any element beta in B_…

Simplified computation of SFT invariants for Legendrian links.

We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive punctures. The construction uses ideas from string topology.

We present an efficient algorithm for calculating the number of components of an integral lamination on an $n$-punctured disk, given its Dynnikov coordinates. The algorithm requires $O(n^2M)$ arithmetic operations, where $M$ is the sum of the absolute values of the Dynnikov coordinates.

We consider constant mean curvature 1 surfaces in $\mathbb{R}^3$ arising via the DPW method from a holomorphic perturbation of the standard Delaunay potential on the punctured disk. Kilian, Rossman and Schmitt have proven that such a surface is asymptotic to a Delaunay surface. We consider families of such potentials p…

We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once-punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs. Our main focus is on the geometr…

Let B_n be the braid group on n > 3 strands. We prove that B_n modulo its center is co-Hopfian. We then show that any injective endomorphism of B_n is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from B_n to B_n+1 is geometric. Additionally, we ob…

In this article, we define and study the total Milnor invariant and the infinitesimal Morita-Milnor homomorphism as punctured disk analogues of the total Johnson map and the infinitesimal Morita homomorphism studied by Kawazumi and Massuyeau in the case of surface of positive genus with one boundary component.

We consider harmonic immersions in $\R^{\N}$ of compact Riemann surfaces with finitely many punctures where the harmonic coordinate functions are given as real parts of meromorphic functions. We prove that such surfaces have finite total Gauss curvature. The contribution of each end is a multiple of $2π$, determined by…

The Dynnikov coordinate system puts global coordinates on the boundary of Teichmüller space of an $n$--punctured disk. We survey the Dynnikov coordinate system, and investigate how we use this coordinate system to study pseudo--Anosov braids making use of results from Thurston's theory on surface homeomorphisms.

We study the convergence of volume forms on a degenerating holomorphic family of log-Calabi-Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let $(X,B)$ be a holomorphic family of sub log canonical, log-Calabi-Yau complex varieties parameterized by the punctured un…

One of the most interesting questions about a group is if its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists and geometers, and is the target of intensive current research. We look at the braid group from a topological point of view (rather …

Let $L\subset J^1(M)$ be a Legendrian submanifold of the 1-jet space of a Riemannian $n$-manifold $M$. A correspondence is established between rigid flow trees in $M$ determined by $L$ and boundary punctured rigid pseudo-holomorphic disks in $T^\ast M$, with boundary on the projection of $L$ and asymptotic to the doubl…

The abstract conjectures a link between knot homologies and quiver partition functions.

We define the Hopf superalgebra U_T sl(1,1), which is a variant of the quantum supergroup U_q sl(1,1), and its tensor product representations V_1^{\otimes n} for n>0. We construct families of DG algebras A, B and R_n, and consider the DG categories DGP(A), DGP(B) and DGP(R_n), which are full DG subcategories of the cat…

In this paper, we compute the graph skein algebra of the punctured disk with two holes. Then, we apply the graph skein techniques developed here to establish necessary conditions for a spatial graph to have a symmetry of order $p$, where $p$ is a prime. The obstruction criteria introduced here extend some results obtai…

New findings on hyperbolicity of augmented links in thickened surfaces.

We present an algorithm for calculating the geometric intersection number of two multicurves on the $n$-punctured disk, taking as input their Dynnikov coordinates. The algorithm has complexity $O(m^2n^4)$, where $m$ is the sum of the absolute values of the Dynnikov coordinates of the two multicurves. The main ingredien…

Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.

The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natu…

Let $S$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $ω$ be a pseudo-Anosov map of $S$ that is obtained from Dehn twists along two families $\{A,B\}$ of simple closed geodesics that fill $S$. Then $ω$ can be realized as an extremal Teichmüller mapping on a surface of type $(p,n)$ which is also denoted by $…

New algebra defined for Legendrian submanifolds, preserving key invariants.

Let $Σ$ be a surface of negative Euler characteristic together with a pants decomposition $¶$. Kra's plumbing construction endows $Σ$ with a projective structure as follows. Replace each pair of pants by a triply punctured sphere and glue, or `plumb', adjacent pants by gluing punctured disk neighbourhoods of the punctu…

In a recent paper A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces $Σ\_n$ in $B^3$ which have genus $0$ and $n$ boundary components, for all $n \geq 3$. For large $n$, we give an independent construction of $Σ\_n$ and prove the existence of free boundary minimal surfaces $\tilde Σ\_n…

The paper studies hyperbolic geometry of links formed by adding trivial components to knots.

In this paper we will present a homological model for Coloured Jones Polynomials. For each colour $N \in \mathbb {N}$, we will describe the invariant $J_N(L,q)$ as a graded intersection pairing of certain homology classes in a covering of the configuration space on the punctured disk. This construction is based on the …