Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.
problem Estimating spectral distribution of twisted Laplacian on hyperbolic surfaces.
method Estimate spectral distribution by supremum norm of harmonic form; show small supremum norm for high genus surfaces; prove uniform Weyl law.
result Prove uniform Weyl law for real parts of spectrum on high genus hyperbolic surfaces.
Study on length distribution of random multicurves on large genus surfaces converging to Poisson-Dirichlet distribution.
problem Length statistics of random multicurves on large genus hyperbolic surfaces.
method Analytical proof of convergence to Poisson-Dirichlet distribution as genus tends to infinity.
result Mean lengths of the three longest components converge to specific percentages of total length as genus increases.
Study finds saddle connections on random surfaces follow Poisson distribution.
problem Distribution of saddle connections on random translation surfaces.
method Analysis of saddle connections on surfaces of large genus.
result Number of saddle connections in given lengths converges to Poisson distribution.
New bounds show triangulated surfaces are evenly distributed in moduli space.
problem Distribution of triangulated surfaces in moduli space as genus increases.
method Proved upper and lower bounds for the number of triangulated surfaces in Teichmüller balls.
result Number of triangulated surfaces in a Teichmüller unit ball is at most exponential in the number of triangles, independent of genus.
The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric. Given suitable restrictions on the genus of the surface, we consider the number…
Random translation surfaces converge to a Poisson plane as genus grows.
problem Understanding the geometric behavior of high genus translation surfaces.
method Proving convergence of random translation surfaces to a Poisson plane using statistical local geometric properties.
result The radius-r neighborhood of a random point in an MSV-distributed random translation surface converges in distribution to the radius r neighborhood of the root in a Poisson translation plane. Random square-tiled surfaces have normal genus distribution and cover all integer vectors.
problem Distribution and properties of random square-tiled surfaces.
method Randomizing model and local central limit theorem for genus.
result The distribution of the genus is asymptotically normal and contains all primitive integer vectors.
This paper studies the critical dynamics of random surfaces, focusing on area and genus evolution.
problem Understanding the time evolution of random surfaces and their genus.
method Analyzes the dynamics of area and genus using Cox-Ingersoll-Ross process and critical phenomena.
result The genus of surfaces evolves into two phases: planar surfaces and foamy surfaces.
In this paper, we study the distribution of the genuses of cluster quivers of finite mutation type. First, we prove that in the 11 exceptional cases, the distribution of genuses is 0 or 1. Next, we consider the relationship between the genus of an oriented surface and that of cluster quivers from this surface. It…
We consider the SO(3) Witten-Reshetikhin-Turaev quantum invariants of random 3-manifolds. When the level r is prime, we show that the asymptotic distribution of the absolute value of these invariants is given by the standard Rayleigh distribution and independent of the choice of level. Hence the probability that the qu…
The paper shows Gaussian fluctuations in eigenvalue statistics of random hyperbolic surfaces.
problem Understanding fluctuations in Laplace eigenvalues of random hyperbolic surfaces.
method Analyzing fluctuations of linear statistics of Laplace eigenvalues over moduli space of surfaces of large genus.
result The distribution of linear statistics tends to a Gaussian as the genus of surfaces increases.
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…
A surface with boundary is randomly generated by gluing polygons along some of their sides. We show that its genus and number of boundary components asymptotically follow a bivariate normal distribution.
Study on geodesics on random hyperbolic surfaces, showing variance asymptotic to X log X.
problem Distribution of closed geodesics on random hyperbolic surfaces.
method Viewing surfaces as random points in moduli space, studying weighted counting function.
result Variance in large genus limit is asymptotic to X log X, with exceptions.
Let Gn be the genus of a two-dimensional surface obtained by gluing, uniformly at random, the sides of an n-gon. Recently Linial and Nowik proved, via an enumerational formula due to Harer and Zagier, that the expected value of Gn is asymptotic to (n−lnn)/2 for n→∞. We prove a local limit theorem…
New knots found with Seifert genus not matching minimal genus Seifert surfaces.
problem Discrepancy between Seifert genus and minimal genus Seifert surfaces.
method Constructed knots with specific genus and handle numbers to demonstrate the discrepancy.
result Found knots where Seifert genus is not realized by minimal genus Seifert surfaces.
The purpose this article is to try to understand the mysterious coincidence between the asymptotic behavior of the volumes of the Moduli Space of closed hyperbolic surfaces of genus g with respect to the Weil-Petersson metric and the asymptotic behavior of the number of arithmetic closed hyperbolic surfaces of genus …
A method to uniformly sample graph-encoded surfaces of fixed size.
problem Sampling uniformly from combinatorial isomorphism types of balanced triangulations of surfaces.
method Relies on connections between graph-encoded surfaces and permutations, and basic properties of the symmetric group.
result Empirical mean genus of the sample is very close to a specific formula as n increases. Construct divide knots with specific genus properties.
problem Understanding the difference between smooth and topological four-genus for knots.
method Construct divide knots with controlled smooth and topological four-genus ratios.
result For strongly quasipositive fibred knots, the ratio between smooth and topological four-genus can be made arbitrarily close to zero.
The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…
The paper generalizes the T-genus to characterize slice knots and slice genus.
problem Characterizing slice knots and slice genus using the T-genus. method Generalizing the T-genus to provide a 3-dimensional characterization of the slice genus. result The difference between the T-genus and the slice genus can be arbitrarily large. Characterizes the Z-genus of boundary links using Blanchfield forms.
problem Determining the Z-genus of boundary links.
method Characterization using Blanchfield forms.
result Variant of shake genus equals Z-genus of knot.
The concordance genus of a knot is the least genus of any knot in its concordance class. Although difficult to compute, it is a useful invariant that highlights the distinction between the three-genus and four-genus. In this paper we define and discuss the stable concordance genus of a knot, which describes the behavio…
Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first 3g−3 geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short (<log(g)) geodesics on a …
Classifies 3-braid knots with maximal 4-genus using McCoy's method.
problem Classifying 3-braid knots with maximal 4-genus.
method McCoy's twisting method and Xu normal form.
result Upper bounds for the topological 4-genus of 3-braid knots.
Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.
problem Exploring differences in non-orientable 4-genus for periodic knots.
method Analyzed p-periodic knots, showing differences in equivariant and non-equivariant non-orientable 4-genus.
result Differences exist in non-equivariant and equivariant non-orientable 4-genus for periodic knots.
Study Khovanov homology of Turaev genus one links, finding a trivial summand.
problem Analyzing Khovanov homology of Turaev genus one links.
method Examined Khovanov homology in specific polynomial gradings.
result Found a trivial summand in Khovanov homology of Turaev genus one links.
Contact connected sums do not increase support genus.
problem Understanding how support genus changes under contact connected sums.
method Analyzing the support genus of contact connected sums of 3-manifolds.
result The support genus of contact connected sums is at most the maximum of the summands' support genera.
The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11--crossing prime knots. This analysis resolves the concordance ge…
This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs, except for in some low complexity cases, all have infinite genus. The curve grap…
Classifies knot traces with specific trisection genus limits.
problem Classifying knot traces with specific trisection genus limits.
method Classifying knot traces with specific trisection genus limits.
result Infinitely many knots have traces with trisection genus 3 and 4, and arbitrarily large trisection genus.
Classifies Teichmüller curves in genus three from genus two.
problem Classifying Teichmüller curves in genus three from genus two.
method Action on homology modulo two of affine groups of translation surfaces.
result Classification of orbits of square-tiled surfaces.
The paper proves trisection genus of Akbulut cork and constructs many corks with trisection genus 3.
problem Determining the trisection genus of contractible 4-manifolds and exotic pairs.
method Proved trisection genus of Akbulut cork and constructed corks with trisection genus 3.
result First examples of contractible 4-manifolds with determined trisection genus except for 4-ball.
New expanders for mean curvature flow contradict genus-reduction conjecture.
problem Contradicting Ilmanen's genus-reduction conjecture for mean curvature flow.
method Construct new expanders asymptotic to cones arising from shrinkers.
result Existence of expanders of arbitrarily large genus.
Study shows systole behavior changes significantly for large genus hyperbolic surfaces.
problem Understanding systole behavior in large genus hyperbolic surfaces.
method Analysis of random surfaces with respect to Weil-Petersson volume.
result Expected value of separating systole behaves like 2logg for large genus. Lower bounds on rational slice genus using Heegaard Floer invariants.
problem Measuring complexity of homology classes in 4-manifolds.
method Introducing rational slice genus, bounding Heegaard Floer τ invariants, using satellite links and closed braids.
result Lower bounds on rational slice genus in terms of Heegaard Floer τ invariants.
Survey of minimal genus problem progress.
problem Finding the smallest genus surface in 3-manifolds.
method Survey and review of existing research.
result Summarizes recent advancements in the minimal genus problem.
Algorithm calculates genus of embedded graphs on surfaces.
problem Finding minimal and maximal genus for graph embeddings.
method Constructing special branched coverings of the 2-sphere.
result Algorithm calculates orientable genus and maximal genus of graphs.
In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both…
For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided…
The paper explores the genus of surfaces in complex projective spaces and improves minimal genus bounds.
problem Investigating the genus of surfaces in complex projective spaces.
method Analyzing knots and torus knots in CP2 and CP2#CP2. result The CP2-genus of knots is unbounded, unlike its topological counterpart. Shows large unknotting number for simple knots.
problem Finding minimum crossing changes for unknotting.
method Positive-to-negative crossing changes without increasing genus.
result Genus non-increasing totally positive unknotting number can be large.
We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fun…
Correspondence found between Askey-Wilson polynomials and genus-two handlebody skein module.
problem Understanding the genus-two handlebody skein module.
method Using q-difference operators for the genus-two skein algebra.
result Correspondence between reduced Askey-Wilson polynomials and genus-two handlebody skein module.
The paper calculates genus bounds for multibranched surfaces.
problem Finding genus bounds for multibranched surfaces.
method Using the first Betti number and boundary genus, the paper provides lower bounds for maximum and minimum genus.
result The maximum and minimum genus of GimesS1 equals twice that of G. The study quantifies distances between certain hyperbolic surfaces and bounds their number.
problem Quantifying distances and bounding numbers of hyperbolic surfaces.
method Parametrization of Teichmüller space by length functions.
result Two coverings of arithmetic surfaces are $P( rac{1}{g})$ apart, with P depending only on the surface. New lower bound for doubly slice genus using knot signatures.
problem Finding a lower bound for the doubly slice genus of knots.
method Using the classical signature function to derive a new lower bound.
result Proved that for every nonnegative integer N, there exists a knot with exactly N difference between slice and doubly slice genus.
In this paper we write explicitly the open book decompositions of links of quotient surface singularities supporting the corresponding unique Milnor fillable contact structure. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the same contact structure. We also investigate wh…