New 3-manifolds with dense ends of specified genus created.
problem Creating 3-manifolds with specific end properties.
method Constructing from Cantor sets in S3; analyzing end genera. result Simply connected 3-manifolds with dense ends of specified genus.
The stable 4-genus of a knot K in 3-space is the limiting value of g_4(nK)/n, where g_4 denotes the 4-genus and n goes to infinity. This induces a seminorm on CQ, the concordance group tensored with the rational numbers. Basic properties of the stable genus are developed, as are examples focused on understanding the un…
This paper presents some finiteness results for the number of boundary slopes of immersed essential surfaces of given genus g in a compact 3-manifold with torus boundary. In the case of hyperbolic 3-manifolds we obtain uniform quadratic bounds in g for the number of possible slopes, independent of the 3-manifold. We al…
We employ cut and paste contact topological techniques to classify some tight contact structures on the closed, oriented genus-2 surface times the interval. A boundary condition is specified so that the Euler class of the of the contact structure vanishes when evaluated on each boundary component. We prove that there e…
New insights into hyperelliptic divisors via integrable Hirota equations.
problem Understanding the geometry of hyperelliptic divisors.
method Proving the vanishing of genus 3 theta constants with even characteristics.
result Integrable Hirota equations specify the structure of hyperelliptic divisors.
Certain classes of 3-manifolds, following Thurston, give rise to a 'skinning map', a self-map of the Teichmüller space of the boundary. This paper examines the skinning map of a 3-manifold M, a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of M to conclude that the skinning map …
Compactifies strata of d-differentials in genus 0.
problem Compactify strata of d-differentials in genus 0.
method Specify an ideal sheaf and show isomorphism to blow-up of M0,n. result Incidence variety compactification is isomorphic to the blow-up of M0,n. Developed Gompf connected sum for orbifolds, constructing symplectic and K-contact manifolds.
problem Constructing symplectic and K-contact manifolds with specific properties.
method Developed Gompf fiber connected sum operation for symplectic orbifolds and used it to construct the required manifolds.
result Constructed a K-contact Smale-Barden manifold with specified 2-homology and sharper estimates.
Let (M,Q) be a compact, three dimensional manifold of strictly negative sectional curvature. Let (Σ,P) be a compact, orientable surface of hyperbolic type (i.e. of genus at least two). Let θ:π1(Σ,P)→π1(M,Q) be a homomorphism. Generalising a recent result of Gallo, Kapovich and Marden concerning necessary and …
The paper provides a criterion for creating Haken manifolds with specific properties.
problem Creating Haken manifolds with prescribed properties.
method Simple criterion and construction methods.
result Construction of many Haken manifolds with specific properties.
This paper reformulates double Hurwitz numbers using topological recursion.
problem Enumerating branched covers with specific ramification over zero and infinity.
method Topological recursion framework applied to double Hurwitz numbers.
result Double Hurwitz numbers are governed by the topological recursion.
Study geometric structure of HCMU surfaces with singularities.
problem Understanding the moduli space of HCMU surfaces with specified genus and conical angles.
method Geometric structure analysis using classical football decomposition and data representation.
result Determined the dimension of the moduli space of HCMU surfaces.
We introduce and study the Hermitian matrix model with potential V(x)=x^2/2-stx/(1-tx), which enumerates the number of linear chord diagrams of fixed genus with specified numbers of backbones generated by s and chords generated by t. For the one-cut solution, the partition function, correlators and free energies are co…
Study geometrically measures to decide if modular companions are conformally equivalent.
problem Deciding if two modular companions are conformally equivalent under a given group action.
method Construct a moduli space and equivariant tilings to measure conformal equivalence.
result Presented a geometric measure to decide conformal equivalence of modular companions.
The paper provides conditions for realizing graphs and polytopes with specified edge lengths.
problem Proving the existence of planar embeddings or polyhedra with specified edge lengths.
method Practical sufficient conditions and software verification for non-self-intersecting perturbations of initial realizations.
result Existence of planar embeddings and polyhedra with specified edge lengths.
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.
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. Paper bounds double slice genus of knots.
problem Understanding knot genus complexities.
method Using Casson-Gordon invariants, the paper defines and bounds the double slice genus.
result Double slice genus can be much larger than slice genus.
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…
Interactive interface for specifying complex tasks through demonstrations.
problem Challenges in specifying high-dimensional reward functions for complex tasks.
method Interactive interface with a set of increasingly complex policies trained on demonstrations.
result Successfully learned a complex task (Lunar Lander) with interactive demonstrations.
Study shows four-genus ratio of two-bridge knots decreases as knots get more complex.
problem Understanding the relationship between smooth four-genus and Seifert genus in two-bridge knots.
method Analytical proof focusing on two-bridge knots and their crossing numbers.
result The expected value of the ratio between smooth four-genus and Seifert genus tends to zero as the crossing number increases.
New examples of local signatures for genus 2 and 3 fibrations.
problem Finding unusual local signatures for specific fibrations.
method Providing specific examples of local signatures for genus 2 and 3 fibrations.
result Examples of local signatures different from usual ones for genus 2 and 3 fibrations.
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.
New link invariant bounds topological slice genus.
problem Bounding the topological slice genus of links.
method Introducing algebraic genus as a new link invariant and using Casson-Gordon invariants.
result Algebraic genus is an upper bound for the topological slice genus.
We specify exactly which groups can act geometrically on CAT(0) spaces whose visual boundary is homeomorphic to either a circle or a suspension of a Cantor set.
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.
A language for specifying complex reinforcement learning tasks.
problem Challenges in specifying and shaping reward functions for complex reinforcement learning tasks.
method Proposes a new language and algorithm for automatically generating and shaping reward functions.
result SPECTRL tool outperforms state-of-the-art baselines.
The paper proves a conjecture about satellite knots and their slice genus.
problem The topological slice genus of satellite knots and its bounds.
method Establishes the conjecture for a variant of the topological slice genus, the Z-slice genus.
result The topological slice genus of a satellite knot is bounded above by the sum of the slice genera of the knot and the pattern.
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.
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.
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. 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.
Classifies changes between genus one fibered knots.
problem Understanding changes between genus one fibered knots.
method Analyzed all monodromies and determined manifolds.
result Classified all generalized crossing changes between genus one fibered knots.
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…
Paper proves non-prime 3-manifolds with specific open book genus have particular prime components.
problem Proving non-prime 3-manifolds with open book genus 2 have non-trivial prime pieces.
method Analyzing open book decompositions and connected sums of 3-manifolds.
result Proves non-prime 3-manifolds with open book genus 2 are homeomorphic to L(p,1)#L(q,1) for some integers p,qeq±1. 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. The paper uses twisting operations to bound knot genus.
problem Bounding the topological slice genus of knots.
method Develops null-homologous twisting operations to study algebraic genus.
result New upper bounds on algebraic genera of torus and satellite knots.