arXiv research

Study on maximal surfaces with high genus in Lorentz-Minkowski space.

Study asymptotics of one part monotone Hurwitz numbers in high genus.

Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.

This paper presents a method to obtain geometric registrations between high-genus ($g\geq 1$) surfaces. Surface registration between simple surfaces, such as simply-connected open surfaces, has been well studied. However, very few works have been carried out for the registration of high-genus surfaces. The high-genus t…

We show that a 3-manifold containing an incompressible surface has topologically minimal surfaces of arbitrary high genus.

The paper constructs surfaces of high genus with three ends.

Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces $ξ_{m,k}$ of high genus by deforming the corresponding DPW potential. As a byproduct, we obtain for fixed $m$ estimates on the area of $ξ_{m,k}$ in terms of their genus $g=m k \gg1$.

We show the existence of infinitely many prime knots each of which having in their complements meridional essential surfaces with two boundary components and arbitrarily high genus.

Study on hyperbolic surfaces' volumes, proving asymptotic expansion for high genus.

The paper finds infinite knots with surfaces of any positive genus.

The study proves surfaces with high genus have a specific inequality.

Large PL surfaces in homology balls can have arbitrarily high genus.

Determines the crossing number of polynomial curve systems on surfaces.

New maps show some surfaces can't be sections of 4D spheres.

In this paper, we define a new metric structure on the shape space of a high genus surface. We introduce a rigorous definition of a shape of a surface and construct a metric based on two energies measuring the area distortion and the angle distortion of a quasiconformal homeomorphism. We show that the energy minimizer …

We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, characterize and simulate networks with a broad range of properties. Remarkably, the study of topologically embedded graphs is non-restrictive because any network can be embedded on a surface with sufficiently high genus. The…

We prove the existence of pure braids with arbitrarily many strands which are small, i.e. they contain no closed incompressible surface in the complement which is not boundary parallel. This implies the existence of irreducible non-Haken 3-manifolds of arbitrarily high Heegaard genus.

In this paper, we calculate the p-torsion of the Farrell cohomology for low genus pure mapping class groups with punctures, where p is an odd prime. Here, `low genus' means g=1,2,3; and `pure mapping class groups with punctures' means the mapping class groups with any number of punctures, where the punctures are not al…

Non-isotopic Heegaard splittings of non-minimal genus were known previously only for very special 3-manifolds. We show in this paper that they are in fact a wide spread phenomenon in 3-manifold theory: We exhibit a large class of knots and manifolds obtained by Dehn surgery on these knots which admit such splittings. M…

We construct a smooth Riemannian metric on any 3-manifold with the property that there are genus zero embedded minimal surfaces of arbitrarily high Morse index.

It is known that there are surface bundles of arbitrarily high genus which have genus two Heegaard splittings. The simplest examples are Seifert fibered spaces with the sphere as a base space, three exceptional fibers and which allow horizontal surfaces. We characterize the monodromy maps of all surface bundles with ge…

Machine learning predicts arithmetic curve invariants with high accuracy.

Study on geodesics on high genus expander surfaces, proving filling and non-simple properties.

An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an $0$-framed 2-h…

Let M and M' be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that M and M' are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is `sufficiently complicated', the Heegaard genus of the amalgamated manifold is completely determine…

We show that the distance of a link $K$ with respect to a bridge surface of any genus determines a lower bound on the genus of essential surfaces and Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries on the knot. In particular, knots with high bridge distance do not admit non-trivial non-hy…

New infinite class of hyperbolic knots with high genus and generalized torsion found.

Kevin Hartshorn showed that if a three-dimensional manifold $M$ admits a Heegaard surface $Σ$ with Hempel distance $d$ then every incompressible surface in $M$ has genus at least $\frac{d}{2}$. Scharlemann-Tomova generalized this, proving that in such a manifold, every other Heegaard surface for $M$ of genus $g' < \fra…

The twisting number of a ribbon knot is at least as large as its doubly slice genus.

The paper constructs families of high genus CMC surfaces in the 3-sphere.

We construct knots in S^3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t,b)-decomposition.

We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative …

We construct families of manifolds that have pairs of genus $g$ Heegaard splittings that must be stabilized roughly $g$ times to become equivalent. We also show that when two unstabilized, boundary-unstabilized Heegaard splittings are amalgamated by a "sufficiently complicated" map, the resulting splitting is unstabili…

Paper finds surface groups can deform in reductive symmetric spaces.

Random translation surfaces converge to a Poisson plane as genus grows.

GOE statistics emerge from surface moduli space averages.

Maps with a single face converge to hyperbolic surfaces in large genus.

If a knot K in a closed, orientable 3-manifold M has a bridge surface T with distance at least 3 in the curve complex of T - K, then the genus of any essential surface in its exterior with non-empty, non-meridional boundary gives rise to an upper bound for the bridge number of K with respect to T. In particular, a nont…

We define a generalization of Coxeter graphs and an associated Coxeter system and Coxeter mapping class. These can be used to construct periodic Coxeter mapping classes on surfaces with arbitrarily large genus, preserving lots of symmetries. The periodic mapping classes can in turn be used to construct sequences of pse…

Path-connectivity of thick laminations on high-genus surfaces.

We show that sub-surfaces of a Heegaard surface for which the relative Hempel distance of the splitting is sufficiently high have to appear in any Heegaard surface of genus bounded by half that distance.

Nonorientable spanning surfaces of periodic knots can have arbitrarily high first Betti number.

Study shows exponential growth of Laplacian determinant on random hyperbolic surfaces.

New minimal surfaces desingularize three Clifford tori, proving uniqueness and characterizing them.

In this paper we study the behavior of the spectrum of a compact, connected Riemannian manifold $(M,g)$ of dimension $d \ge 2$, when we add an increasing number of increasingly small handles. No assumptions on any of the curvatures are needed.

Exact diameter found for some Riemann surfaces.

Study shows energy levels on hyperbolic surfaces follow GOE fluctuations.

We show that there are hyperbolic tunnel-number one knots with arbitrarily high bridge number and that "most" tunnel-number one knots are not one-bridge with respect to an unknotted torus. The proof relies on a connection between bridge number and a certain distance in the curve complex of a genus-two surface.