Prove integrality of genus-g indices with adjoint Reidemeister torsions for twist knots and meridians.
problem Prove integrality of genus-g indices with adjoint Reidemeister torsions for twist knots and meridians. method Consider the sum of the adjoint Reidemeister torsions and prove integrality for twist knots and meridians.
result Prove integrality of genus-g indices with adjoint Reidemeister torsions for twist knots and meridians. New minimal surfaces found in 3D sphere with low genus.
problem Finding minimal surfaces in 3D sphere with low genus.
method Equivariant min-max procedure applied to a novel sweepout.
result Discovered new minimal surfaces with low genus.
Arithmetic Kontsevich-Zorich monodromy found in a specific origami surface.
problem Exploring the monodromy of a symmetric origami in genus 4.
method Analyzing the Veech group and symplectic group properties of the origami.
result Existence of arithmetic Kontsevich-Zorich monodromy in a specific origami.
Let M be a Spin-manifold with S1-action and let σ∈S1 be of finite order. We show that the indices of certain twisted Dirac operators vanish if the action of σ has sufficiently large fixed point codimension. These indices occur in the Fourier expansion of the elliptic genus of M in one of its cusps. As …
New gauge invariants from framed 3-manifolds match Hopf algebra indicators.
problem Computing indicators for Hopf algebras.
method Using Kuperberg invariants from framed 3-manifolds.
result Kuperberg invariants match higher Frobenius-Schur indicators of Hopf algebras.
Several independent articles have observed that the Hirzebruch χy-genus has an important feature, which the author calls -1-phenomenon and tells us that the coefficients of the Taylor expansion of the χy-genus at y=−1 have explicit expressions. Hirzebruch's original χy-genus can be extended towards two direc…
Libgober and Wood proved that the Chern number c1cn−1 of a n-dimensional compact complex manifold can be determined by its Hirzebruch χy-genus. Inspired by the idea of their proof, we show that, for compact, spin, almost-complex manifolds, more Chern numbers can be determined by the indices of some twist…
Simon's knot genus problem solved with 3-manifold groups.
problem If epimorphism exists between knot groups, knot genus of one is greater than or equal to the other.
method Proved conjecture linking 3-manifold groups and Thurston norms, showed locally indicable groups are Lewin groups.
result Existence of epimorphism between knot groups implies knot genus inequality.
Efficient multisections found for odd-dimensional tori.
problem Finding optimal multisections of odd-dimensional tori.
method Constructing multisections with specific properties (genus n, symmetry) for odd-dimensional tori. result Optimal multisections of genus n found for odd-dimensional tori. The paper improves bounds on geodesic lengths and their simplicity on hyperbolic surfaces.
problem Estimating the distribution of geodesic lengths on hyperbolic surfaces.
method Analyzing the moduli space of hyperbolic surfaces with the Weil-Petersson metric.
result Most geodesics of certain lengths are simple and non-separating, confirming a conjecture.
Paper computes knot symmetric quandle for surface-links and finds infinitely many distinct surface-knots.
problem Computing knot symmetric quandle for surface-links.
method Using plat form presentations, the paper computes the knot symmetric quandle for surface-links.
result Infinitely many distinct surface-knots of genus g with plat indices m.
Study on constant mean curvature 1-immersions into hyperbolic 3-manifolds.
problem Understanding CMC 1-immersions of surfaces into hyperbolic 3-manifolds.
method Parametrization of moduli space, analysis of blow-up phenomena, asymptotic analysis.
result Sharp condition for genus g=2 involving Kodaira map at six Weierstrass points.
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 …
New methods classify convex lattice polygons for affine dimers.
problem Not all convex lattice polygons are characteristic polygons of affine dimers.
method General constructions and algorithm for finding affine dimers with prescribed polygons.
result All lattice triangles, generalised parallelograms, and polygons of genus at most two admit an affine dimer.
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…
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.
We exhibit geometric situations, where higher indices of the spinor Dirac operator on a spin manifold N are obstructions to positive scalar curvature on an ambient manifold M that contains N as a submanifold. In the main result of this note, we show that the Rosenberg index of N is an obstruction to positive sc…
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…
Interpretable neural network for plant traits and species identification.
problem Plant phenotyping and identification.
method Neural network trained on UPWINS spectral library, with visualization of weights for trait-based spectral features.
result 90% accuracy in species identification with interpretable neural network.
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.
The conformal parameterisation of a minimal surface is harmonic. Therefore, a minimal surface is a critical point of both the energy functional and the area functional. In this paper, we compare the Morse index of a minimal surface as a critical point of the area functional with its Morse index as a critical point of t…
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.
The paper studies 3-manifolds with specific Morse-Smale diffeomorphisms and finds they are homeomorphic to lens spaces.
problem Understanding the topology of 3-manifolds with certain Morse-Smale diffeomorphisms.
method Analyzing the structure of fixed points and separatrices of diffeomorphisms in 3-manifolds.
result All supporting manifolds of these diffeomorphisms are homeomorphic to lens spaces.
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.
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.
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…