arXiv research

Formula proves Euler characteristic of singularized surfaces.

We provide a classification of the essential surfaces of non-negative Euler characteristic in the exteriors of genus two handlebodies embedded in the 3-sphere.

Study Euler class of surface bundles with nontrivial results.

Researchers refine the non-orientable $4$-genus of torus knots using Batson's surfaces.

This paper classifies semi-equivelar maps on a special surface.

Formula for genus of 3D flows based on boundary data.

NP-hard problem found for non-orientable surfaces in 3D manifolds.

New bounds on Seifert surfaces for alternating links are found.

Study of closed real plane curves with hyperelliptic genus three solutions.

We show that when the genus and punctures of a surface are directly proportional by some rational number the minimal asymptotic translation length in the curve complex has behavior inverse to the square of the Euler characteristic. We also show that when the genus is fixed and the number of punctures varies the behavio…

We calculate the Masur-Veech volume of gothic Teichmüller curves in genus four.

The paper shows Euler classes for homeomorphisms of Seifert fibered 3-manifolds are unbounded.

Essential embeddings of metric graphs on hyperbolic surfaces are constructed and studied.

We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmu…

Study counts specific surfaces in Montesinos knots with 4 rational tangles.

Let $e$ denote the Euler class on the space $Hom(Γ_g, PSL(2,\mathbb R))$ of representations of the fundamental group $Γ_g$ of the closed surface $Σ_g$ of genus $g$. Goldman showed that the connected components of $Hom(Γ_g, PSL(2,\mathbb R))$ are precisely the inverse images $e^{-1}(k)$, for $2-2g\leq k\leq 2g-2$, and t…

Researchers compute dimensions of GLN-skein modules for genus-one mapping tori.

Study on the Euler-Plateau energy with elastic modulus, focusing on minimizers and critical surfaces.

We explore the dynamics of the action of the mapping class group in genus 2 on the PSL(2,R)-character variety. We prove that this action is ergodic on the connected components of Euler class 1 and -1, as it was conjectured by Goldman. In the connected component of Euler class 0 there are two invariant open subsets, on …

The paper shows how to unknot certain nonorientable surfaces in 4-dimensional spaces.

New 4-manifolds meet trisection bounds.

New vanishing theorems for genera derived under almost nonnegative Ricci curvature.

New proof of Milnor-Wood inequality for circle bundles.

We investigate the bounded cohomology of Lefschetz fibrations. If a Lefschetz fibration has regular fiber of genus at least 2 and it has at least two distinct vanishing cycles, we show that its Euler class is not bounded. As a consequence, we exclude the existence of negatively curved metrics on Lefschetz fibrations wi…

Study Euler characteristic of manifolds with almost nonnegative curvature operator, showing nonnegativity under certain conditions.

Minimal genus surfaces solve homology problems in finite complexes.

New proof of homological stability for surface mapping classes.

We construct sequences of pseudo-Anosov mapping classes whose dilatations behave asymptotically like the inverse of the Euler characteristic of the surface they are defined on. These sequences are used to show that if the genus, g, and punctures, n, of a surface are related by a rational ray g=rn then the minimal dilat…

The paper studies spectral properties of Jacobi operator for surfaces with nonpositive Euler characteristic.

Classifies 4-manifolds with genus one horizontal handlebody decomposition.

Given a 3-manifold M containing an incompressible surface Q, we obtain an inequality relating the Heegaard genus of M and the Heegaard genera of the components of M - Q. Here the sum of the genera of the components of M - Q is bounded above by a linear expression in terms of the genus of M, the Euler characteristic of …

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 …

In this note we define three invariants of contact structures in terms of open books supporting the contact structures. These invariants are the support genus (which is the minimal genus of a page of a supporting open book for the contact structure), the binding number (which is the minimal number of binding components…

The paper explores actions of surface mapping class groups on 3-manifolds.

Surface bundles' signatures tied to their Euler characteristics.

For the class of quasi-periodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction and the geometry of the evolving curves. We give a complete description of genus one solutions, including geometrically interesting special cases such a…

It is a conjecture that the signature of a positive link is bounded below by an increasing function of its negated Euler characteristic. In relation to this conjecture, we apply the generator description for canonical genus to show that the boundedness of the genera of positive knots with given signature can be algorit…

Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.

We study groups of C^1 orientation-preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly-orderable groups. We show that every such group with a bounded orbit is circularly-orderable, and show that certain generalized braid groups are circularly-orderable. We also show that the …

Counted essential surfaces in a knot's exterior, finding a unique pattern.

Counterexamples to a conjecture on ribbon graph genus changes were found and proven.

The paper defines new modular forms from almost complex manifolds and derives anomaly cancellation formulas.

Origami graphs' Euler characteristics grow as origami complexity increases.

In this paper, we prove that we can recover the genus of a closed compact surface $S$ in $\mathbb{R}^3$ from the restriction to a generic line of the Fourier transform of the canonical measure carried by $S$. We also show that the restriction on some line in Minkowski space of the solution of a linear wave equation who…

Researchers found two types of graphs for 6D torus manifolds with Euler number 6.

Let K' be a hyperbolic knot in S^3 and suppose that some Dehn surgery on K' with distance at least 3 from the meridian yields a 3-manifold M of Heegaard genus 2. We show that if M does not contain an embedded Dyck's surface (the closed non-orientable surface of Euler characteristic -1), then the knot dual to the surger…

We investigate constraints on embeddings of a non-orientable surface in a $4$-manifold with the homology of $M \times I$, where $M$ is a rational homology $3$-sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsváth--Sazbó $d$-invariants or …

We exhibit an infinite family of tight contact structures with the property that none of the supporting open books minimizes the genus and maximizes the Euler characteristic of the page simultaneously, answering a question of Baldwin and Etnyre in arXiv:0910.5021 .