Study maximal orders to find surfaces without genus 212.
problem Which genera can be obtained by closed congruence surfaces?
method Investigate maximal orders in quaternion algebras to construct surfaces.
result No surface of genus 212 can be constructed by maximal orders.
We show that the distance trisector curve is not an algebraic curve, as was conjectured in the founding paper by T. Asano, J. Matousek and T. Tokoyama: "The distance trisector curve", Advances in Math., 212, 338-360 (2007).
New KWS neural networks improve accuracy and power efficiency.
problem Maximizing accuracy and power efficiency in KWS systems.
method Hardware Aware Training (HAT) for LMU-based neural networks.
result Achieved state-of-the-art accuracy and low parameter counts.
Study finds mixed evidence of monthly stock market anomalies in Turkey and US.
problem Investigating whether stock markets exhibit abnormal returns monthly.
method Statistical summary analysis, decomposition technique, dummy variable estimation, binary logistic regression.
result Weak evidence against efficient market hypothesis on monthly returns, with notable May effect in Turkey.
Machine learning models for COVID-19 detection and prognosis from chest images are flawed and unreliable.
problem Developing reliable machine learning models for COVID-19 diagnosis and prognosis from chest images.
method Systematic review of machine learning models published in 2020.
result None of the models identified are of clinical use due to methodological flaws and biases.
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. 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…
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.
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.
Deep learning ranks response surfaces for optimal stopping problems in finance.
problem Ranking response surfaces in stochastic control problems.
method Reformulate as image segmentation problem and apply deep learning algorithms.
result Deep learning provides an efficient method for solving optimal stopping problems.
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.
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.
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. 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.
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.
A new method shows open Riemann surfaces have finite genus.
problem Determining the genus of open Riemann surfaces.
method Introducing a quotient space of forms to determine finite genus.
result Open Riemann surfaces have finite genus and can be embedded in a compact surface.
The paper studies Turaev genus one links and finds their Khovanov homology to be isomorphic to Z in at least one extremal grading.
problem Understanding the extremal properties of Turaev genus one links.
method Analyzing Khovanov homology of Turaev genus one links.
result Khovanov homology of Turaev genus one links is isomorphic to Z in at least one extremal grading.
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…
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. 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.
Study cohomology of mapping class groups for big genus surfaces.
problem Characterize the first cohomology of pure mapping class groups for genus one and zero surfaces.
method Analyzes the first integral cohomology of pure mapping class groups for infinite type genus one surfaces and genus zero surfaces.
result For genus one surfaces, the first integral cohomology is trivial. For genus zero surfaces, there is an uncountable family of non-trivial homomorphisms to Z.
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.
Compact self-shrinkers in 3D with fixed genus and entropy.
problem Bounding the number of ends of self-shrinkers.
method Proving compactness with bounded entropy and fixed genus.
result Uniform bounds on the number of ends by entropy and genus.
The study establishes a link between the complexity of fibered knots and the genus of their Heegaard splittings.
problem Understanding the complexity of Heegaard splittings induced by fibered knots.
method Analyzing the monodromy of fibered knots and their impact on Heegaard splittings.
result Minimal genus Heegaard splittings of a three-manifold are unique and can be induced by fibered knots with complex monodromies.
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…
An open book decomposition of a 3-manifold M induces a Heegaard splitting for M, and the minimal genus among all Heegaard splittings induced by open book decompositions is called the \emph{open book genus} of M. It is conjectured by Ozbagci \cite{O} that the open book genus is additive under the connected sum of …
It is well-known that Heegaard genus is additive under connected sum of 3-manifolds. We show that Heegaard genus of contact 3-manifolds is not necessarily additive under contact connected sum. We also prove some basic properties of the contact genus (a.k.a. open book genus) of 3-manifolds, and compute this invariant fo…
Study knots with genus one, finds Gordian distance and cosmetic crossing constraints.
problem Understanding knots with genus one and their properties.
method Using HOMFLT polynomials to find obstructions for Gordian distance and cosmetic crossings.
result Proves the (generalized) cosmetic crossing conjecture for genus one pretzel knots.
New bounds on knot complexity defined via band number and surface diagrams.
problem Understanding the complexity of knots and their generalizations.
method Defining and analyzing band number, using broken surface diagrams and 3-manifold embeddings.
result Band number is an upper bound on double slice genus, a knot complexity measure.
For d≥2, the regular genus of a closed connected PL d-manifold M is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of M imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every…