Researchers refine the non-orientable 4-genus of torus knots using Batson's surfaces.
problem Finding the minimum non-orientable 4-genus for torus knots. method Developed and analyzed Batson's non-orientable spanning surfaces in B4. result Batson's surfaces minimize the non-orientable 4-genus among certain surfaces. We found an infinite family of counterexamples to Batson's conjecture.
problem Batson's conjecture about nonorientable 4-ball genus of torus knots is false.
method We identified an infinite family of counterexamples to Batson's conjecture.
result We found an infinite family of counterexamples to Batson's conjecture.
We show that the torus knot T4,9 bounds a smooth Möbius band in the 4-ball, giving a counterexample to Batson's non-orientable analogue of Milnor's conjecture on the smooth slice genera of torus knots.
For a given knot, we study the minimal number of positive eigenvalues of the double branched cover over spanning surfaces for the knot. The value gives a lower bound for various genera, the dealternating number and the alternation number of knots, and we prove that Batson's bound for the non-orientable 4-genus gives an…
We compute the non-orientable 4-ball genus for a new family of torus knots.
problem Computing the non-orientable 4-ball genus for a new family of torus knots.
method Combination of band surgeries and known bounds.
result Computed the non-orientable 4-ball genus for the knots in the family.
Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.
problem Understanding non-orientable surfaces and their inscribed rectangles.
method Analyzing smooth and locally-flat non-orientable surfaces in 4-ball with specific knots, comparing results.
result Established differences between smooth and locally-flat non-orientable 4-genus of torus knots.
The nonorientable 4-genus γ4(K) of a knot K is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot K. We study a conjecture proposed by Batson about the value of γ4 for torus knots, which can be seen as a nonorientable analogue of Milnor's Conjec…
In this paper, we provide a novel construction of the linear-sized spectral sparsifiers of Batson, Spielman and Srivastava [BSS14]. While previous constructions required Ω(n4) running time [BSS14, Zou12], our sparsification routine can be implemented in almost-quadratic running time O(n2+ε). The funda…
Geography problem for nonorientable surfaces bounded by knots.
problem Bounding and computing the nonorientable 4-genus of knots.
method Analysis of existing methods, relationships between Betti number and normal Euler class, exploration of families of torus knots, use of Ozsváth-Szabó d-invariant.
result Improvement on the bound for some knots using the Upsilon invariant.
We apply the Rasmussen spectral sequence to prove that the Z3-graded vector space structure of the HOMFLYPT homology over Z2 detects unlinks. Our proof relies on a theorem of Batson and Seed stating that the Z2-graded vector space structure of the Khovanov homology over $\mathbb{Z}_2…
For a closed 4-manifold X and a knot K in the boundary of punctured X, we define γX0(K) to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured X with boundary K. Note that γS40 is equal to the non-orientable 4-ball genus and hence γX0 is a generalizati…
Classifies links with small Khovanov homology ranks.
problem Classifying links with specific ranks in Khovanov homology.
method Previous results combined with new classifications.
result All links with ranks ≤ 8 and three-component links with ranks ≤ 12 are classified.
If L is an oriented link with n components, then the rank of its Khovanov homology is at least 2n. We classify all the links whose Khovanov homology with Z/2-coefficients achieves this lower bound, and show that such links can be obtained by iterated connected sums and disjoint unions of Hopf links and unknots. Th…
New bounds on nonorientable four-ball genus for torus knots.
problem Finding bounds on the nonorientable four-ball genus of torus knots.
method Combining knot Floer homology techniques to derive lower bounds.
result Sharp bounds for several families of torus knots, including T4n,(2n±1)2. The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the sp…
Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson-Seed arX…
By considering negative surgeries on a knot K in S3, we derive a lower bound to the non-orientable slice genus γ4(K) in terms of the signature σ(K) and the concordance invariants Vi(K), which strengthens a previous bound given by Batson, and which coincides with Ozsváth-Stipsicz-Szabó's bound in…
Study shows links with unbounded annular Khovanov gradings but bounded Floer gradings.
problem Understanding gradings in annular Khovanov homology.
method Infinite families of annular links, computations, satellite operations, link splitting spectral sequence.
result Existence of links with unbounded annular Khovanov gradings but bounded Floer gradings.
Paper shows that for torus knots, the pinch number equals the unoriented band unknotting number.
problem Determining the minimum number of band surgeries to unknot torus knots.
method Used the torsion order of unoriented knot Floer homology.
result Pinch number and unoriented band unknotting number coincide for torus knots.
We define a deformation of the triply graded Khovanov-Rozansky homology of a link L depending on a choice of parameters yc for each component of L, which satisfies link-splitting properties similar to the Batson-Seed invariant. Keeping the yc as formal variables yields a link homology valued in triply graded …
The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split link. We obtain a lower bound on the splitting number in terms of the (multivariable) signature and nullity. Although very elementary and easy to compute, this bound turns out to be …
The paper classifies links with low rank knot Floer and Khovanov homologies.
problem Detecting and classifying links with low rank knot Floer and Khovanov homologies.
method Generalized link Floer homology, used to obtain rank bounds and classify links.
result Knot Floer homology detects T(2,8) and T(2,10). Study of cuspidal edges on focal surfaces of regular surfaces.
problem Clarifying the sign of singular curvature at cuspidal edges.
method Investigation using singularities of parallel surfaces.
result Clarification of the sign of singular curvature at cuspidal edges.
New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.
problem Limiting the outcomes of gluing Scherk surfaces into minimal surfaces.
method Constructing minimal surfaces by stacking and gluing doubly periodic Scherk surfaces.
result Except for special cases, gluing more Scherk surfaces results in known minimal surfaces.
Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.
problem Investigate differential geometry properties of focal surfaces of lightcone framed surfaces.
method Introduced lightcone frame to define lightcone framed surfaces, then investigated their differential geometry properties.
result Investigated differential geometry properties of focal surfaces of lightcone framed surfaces.
Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
The paper studies special surfaces with a new type of support function.
problem Characterizing surfaces with a specific quadratic support function.
method Developed a Weierstrass type representation involving holomorphic functions.
result Classified surfaces of rotation with this new type of support function.
The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.
problem Understanding the isotopy and closure properties of knitted surfaces and surface-links.
method Analyzing the structure and closure of knitted surfaces and surface-links in R4. result Any surface-link is ambient isotopic to the closure of a 2-dimensional knit.
We investigate surfaces with constant harmonic-mean curvature one (HMC-1 surfaces) in hyperbolic three-space. We allow them to have certain kinds of singularities, and discuss some global properties. As well as flat surfaces and surfaces with constant mean curvature one (CMC-1 surfaces), HMC-1 surfaces belong to a cert…
Unstable minimal surfaces in n-space link to hyperbolic products.
problem Characterizing unstable minimal surfaces in Rn and their product counterparts. method Lifting to R-trees, deforming to hyperbolic products, and proving instability equivalence. result Unstable minimal surfaces in Rn imply unstable surfaces in product hyperbolic spaces. Researchers generalize Ribaucour-type surfaces with new mathematical representation.
problem Defining and characterizing new geometric surfaces.
method Developed a new mathematical representation for GRT-surfaces involving holomorphic functions and a real function.
result Explicit examples and classification of GRT-surfaces of rotation.
Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.
problem Understanding automorphisms of fine curve graphs on surfaces.
method Analyzing vertices and edges of fine curve graphs to match with surface homeomorphisms.
result Automorphism group of fine curve graphs is naturally isomorphic to the homeomorphism group of boundaryless planar surfaces with at least 7 punctures.
This paper connects Laguerre minimal surfaces to Weierstrass representations.
problem Understanding the relationship between Laguerre minimal surfaces and Weierstrass representations.
method Defining spherical mean curvature and providing Weierstrass-type representations for two classes of surfaces.
result Laguerre minimal surfaces are related to H2-surfaces, providing a new Weierstrass-type representation. New surfaces in 4-ball constructed from knits, described by charts.
problem Constructing surfaces in 4-ball from knits.
method Introducing knitted surfaces, describing them with BMW charts.
result Every compact surface in 4-ball is ambiently isotopic to a knitted surface.
Study on singular points of translation surfaces under linearly dependent conditions.
problem Investigate singular points of translation surfaces under linearly dependent conditions.
method Use theories of generalised framed surfaces and framed surfaces.
result Introduce translation generalised framed surfaces and investigate their singular points.
Crochet patterns for minimal surfaces created using trigonometry.
problem Creating crochet patterns for minimal surfaces.
method Using trigonometric identities to calculate arc lengths.
result Crochet instructions for Enneper's surface.
New surfaces generalize Dini surfaces in 4D.
problem None explicitly stated; focuses on surface generalization.
method Introducing a new family of surfaces in 4D.
result Generalized Dini surfaces exist in 4D.
Here, we focus on focal surfaces of a tubular surface in Euclidean 3-space E^3: Firstly, we give the tubular surfaces with respect to Frenet and Darboux frames. Then, we define focal surfaces of these tubular surfaces. We get some results for these types of surfaces to become flat and we show that there is no minimal f…
New method classifies HCMU surfaces in 3D space forms as Weingarten surfaces.
problem Classifying HCMU surfaces in 3D space forms as Weingarten surfaces.
method Totally different method from previous work.
result Criteria for Weingarten surfaces that are also HCMU surfaces.
Minimal surfaces are the only biharmonic in Sol3.
problem Characterizing biharmonic surfaces in Sol3.
method Found local equations for biconservative surfaces and showed all biharmonic surfaces are minimal.
result All biharmonic surfaces in Sol3 are minimal.
It is shown that any handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link up to equivalences, so that every stable-ribbon surface-link is a ribbon surface-link. This is a generalization of a previously observed result for a stably trivial surface-link. Two observations are give…
This paper proves compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
problem Compactness of conformal Chern-minimal surfaces in Hermitian surfaces.
method Proves compactness through bubble tree limit analysis.
result Compactness of conformal Chern-minimal surfaces is established with bounded area.
Study proves surfaces with constant curvature are simple shapes.
problem Characterizing singular minimal surfaces with constant curvature.
method Proved geometric properties of surfaces with constant curvature.
result Singular minimal surfaces with constant curvature are planes, spheres, and cylindrical surfaces.
New surface class defined using osculating circles.
problem Defining a new surface class in Euclidean space.
method Using osculating circles of curves and classification of specific types.
result Classification of canal and Weingarten surfaces.
The article constructs Bolza-like surfaces for infinitely many genera and studies their properties.
problem Maximizing systole functions in Teichmüller spaces for genus two and higher.
method Defining and constructing Bolza-like surfaces with specific triangulations and properties.
result Global maximal surfaces can be constructed using Bolza-like surfaces, and systolic geodesics intersect at even points.
The Enneper surface and helix surfaces are unique in their geometric properties.
problem Characterizing surfaces with specific geometric properties.
method Analyzing isogonal lines and pseudo-geodesic lines in 3D Euclidean space.
result Helix surfaces and Enneper surface are the only surfaces with isogonal lines as generalized helices and pseudo-geodesic lines.
Study on automorphisms of K3 and Enriques surfaces, proving entropy gaps and achirality.
problem Entropy norms and achirality of automorphisms on K3 and Enriques surfaces.
method Proves gap theorems for entropy norms and studies achirality in terms of genus-one fibrations.
result Entropy gaps and achirality results for automorphisms of K3 and Enriques surfaces.
We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the…