The paper studies cylindrical handlebody-knots of genus two with unique unknotting annuli and finds trivial symmetry groups.
problem Understanding the topology and symmetry of cylindrical handlebody-knots of genus two.
method Analysis of Thurston's hyperbolization theorem and investigation of unknotting annuli.
result The symmetry group is trivial if the unknotting annulus is unique and of type 2. The only knots that are tunnel number one and genus one are those that are already known: 2-bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Munoz and by Morimoto-Sakuma. This confirms a conjecture first made by Goda and Teragaito.
Proves a plumbing-multiplicative property of a Links-Gould invariant.
problem Proving a multiplicative property of the Links-Gould invariant.
method Using Laurent polynomials and topological plumbing of surfaces.
result The Links-Gould invariant is monic in the Laurent polynomial ring.
Any knot in S3 may be reduced to a slice knot by crossing changes. Indeed, this slice knot can be taken to be the unknot. In this paper we study the question of when the same holds for knots in homology spheres. We show that a knot in a homology sphere is nullhomotopic in a smooth homology ball if and only if that k…
Classifies essential annuli in a genus two handlebody exterior.
problem Classifying essential annuli in a genus two handlebody exterior.
method Building on JSJ-graph classification and essential annuli classification.
result Characterizes the numbers of different types of essential annuli in an infinite family.
Study minimal annuli in a slab, estimating their area.
problem Estimating the area of minimal annuli in a slab.
method Organized minimal annuli based on winding number, deduced convexity of length function, compared to catenoid waist area.
result Deduced convexity of length function and estimated area of minimal annuli.
The study constructs free boundary CMC annuli in spherical and hyperbolic balls.
problem Finding free boundary CMC annuli in spherical and hyperbolic balls.
method Constructing free boundary CMC annuli with constant mean curvature H in geodesic balls of S^3 and H^3.
result Embedded free boundary CMC annuli exist for certain mean curvatures in both spaces.
Sharp lower bound found for area of vector fields on spherical annuli.
problem Finding the minimum area of unit vector fields on spherical annuli.
method Established a sharp lower bound through mathematical analysis.
result Sharp lower bound for the area of unit vector fields on spherical annuli.
We define a new notion of thin position for a graph in a 3-manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the idea of thin position for knots first originated by Gabai. This thin position has the property that connect summing annuli and pairs-of-pants …
Minimal annuli constructed in PSL2 via variational method.
problem Constructing minimal annuli in a non-symmetric 3-manifold.
method Variational method, foliations by minimal surfaces, limit of compact minimal annuli.
result Existence of complete, embedded minimal annuli asymptotic to vertical planes.
In S2×R there is a two-parameter family of properly embedded minimal annuli foliated by circles. In this paper we show that this family contains all properly embedded minimal annuli. We use the description of minimal annuli in S2×R by periodic harmonic maps $G : \…
New minimal annuli found in unit ball, solving old problems.
problem Constructing free boundary minimal annuli in unit ball.
method Symmetric and foliated by spherical curvature lines.
result First non-embedded free boundary minimal annuli in unit ball.
New minimal discs and annuli found in ellipsoids.
problem Constructing minimal surfaces in ellipsoids.
method Equivariant variational methods.
result At least three distinct embedded free boundary minimal annuli in ellipsoids.
We prove that maximal annuli in L3 bounded by circles, straight lines or cone points in a pair of parallel spacelike planes are part of either a Lorentzian catenoid or a Lorentzian Riemann's example. We show that under the same boundary condition, the same conclusion holds even when the maximal annuli hav…
Classifies essential annuli in genus two handlebody-knots, determining hyperbolicity and constructing obstructions.
problem Classifying essential annuli in genus two handlebody-knots.
method Introducing τ- and ρ-tangles and good rectangles, classifying these structures.
result Categorization of atoroidal 3-decomposable genus two handlebody-knots based on essential annuli.
Improved flatness in annuli using PDE methods.
problem Flatness improvement in annuli.
method PDE-based approach adapted to exterior domains.
result Alternative proof of minimal surface end-structure and asymptotics.
Study eigenvalues and eigenfunctions of fourth-order operators in annuli, proving optimal estimates and non-radiality.
problem Eigenvalue and eigenfunction analysis of fourth-order operators in degenerating annuli.
method Optimal estimates and non-radiality results for eigenfunctions in annuli.
result Nigh optimal estimate for the first eigenvalue and non-radiality of eigenfunctions in degenerating annuli.
Constructs minimal annuli with free boundary in hyperbolic 3-space.
problem Finding minimal surfaces with boundary in hyperbolic geometry.
method Constructs families of non-rotational minimal annuli with shared symmetry.
result Bifurcates from hyperbolic catenoids, forming a countable collection.
Existence of minimal annuli in 3-sphere with boundary on geodesic spheres.
problem Existence of free boundary minimal annuli in 3-sphere.
method One-parameter family of complete minimal immersions of R × S^1 into S^3, analysis of Otsuki tori.
result Existence of embedded free boundary minimal annuli contained in geodesic balls.
We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil_3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil_3 is entire. Also, the si…
We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature − including strictly convex domains of the Euclidean space R3.
The paper proves a statement about surfaces diffeomorphic to annuli.
problem Proving a statement about surfaces diffeomorphic to annuli in Perelman's paper.
method Uses extrinsic techniques, co-area formula, and is potentially generalizable.
result Potential generalizability to higher dimensions.
Brunnian theta curves in 3D spheres have hyperbolic exteriors.
problem Characterizing Brunnian theta curves in 3D spheres.
method Sutured manifold theory, classification of annuli, and Thurston's hyperbolic geometry.
result Brunnian theta curves of low bridge number have hyperbolic exteriors with totally geodesic boundary.
A semigroup of annuli integrates a central extension of vector fields on S^1.
problem No Lie group exists for complexified vector fields on S^1.
method Introduced an enlargement of the semigroup of annuli and proved it integrates a central extension of vector fields.
result Every partially thin annulus is the time-ordered exponential of a path in the cone of inward pointing complexified vector fields.
Constructs minimal surfaces near the boundary of a ball.
problem Creating minimal surfaces close to the boundary of a ball.
method PDE gluing methods to construct FBMS of genus zero.
result Desingularizations of catenoidal annuli and flat discs near the boundary.
We explicitly classify all S1-invariant free boundary minimal annuli and Möbius bands in Bn. This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for S1-invariant metrics on the annulus and Möbius band. First, we determine the supremum of the k-th normaliz…
Alexander polynomial derived from knot contact homology and Floer strips.
problem Calculating the Alexander polynomial of a knot.
method Contact homology and Floer theory applied to knot complements.
result Alexander polynomial expressed as an integral of partial derivatives.
New examples show non-rotational annuli in a ball, solving a uniqueness problem.
problem Uniqueness of annular solutions in a ball.
method Constructing a family of compact embedded CMC annuli with free boundary in the unit ball.
result Non-rotational annuli found, providing a counterexample to Nitsche and Wente's uniqueness problem.
In previous work with Schoenfeld, we considered a string-type chain complex of curves on surfaces, with differential given by resolving crossings, and computed the homology of this complex for discs. In this paper we consider the corresponding "string homology" of annuli. We find this homology has a rich algebraic stru…
We show that after stabilizations of opposite parity and braid isotopy, any two braids in the same topological link type cobound embedded annuli. We use this to prove the generalized Jones conjecture relating the braid index and algebraic length of closed braids within a link type, following a reformulation of the prob…
We study minimal annuli in S2×R of finite type by relating them to harmonic maps C→S2 of finite type. We rephrase an iteration by Pinkall-Sterling in terms of polynomial Killing fields. We discuss spectral curves, spectral data and the geometry of the isospectral set…
Study links with annuli using sutured Floer homology.
problem Characterize links with specific cable structures.
method Apply sutured Floer homology techniques.
result Characterizations of links with (n,nm)-cables and (2,2m)-cables. In this paper we study the moduli space of properly Alexandrov-embedded, minimal annuli in H2×R with horizontal ends. We say that the ends are horizontal when they are graphs of C2,α functions over ∂∞H2. Contrary to expectation, we show that one can …
Study proves all free boundary CMC annuli are of finite type.
problem Free boundary constant mean curvature annuli in the unit ball.
method Adapted Sklyanin's K-matrix formalism to sinh-Gordon equation.
result All free boundary CMC annuli are of finite type.
The critical catenoid is uniquely determined by certain symmetries of its boundary.
problem Uniqueness of free boundary minimal annuli in a half-ball.
method Symmetry analysis and boundary conditions.
result An embedded free boundary minimal annulus with specific symmetries is congruent to the critical catenoid.
We show a non existence result for solutions of the prescribed mean curvature equation in the product manifold H2×R, where H2 is the real hyperbolic plane. More precisely we prove a-priori estimates for graphs with constant mean curvature h∈(0,1/2] on circular annuli of $\mathbb{H…
The study finds many Möbius bands and annuli on toroids.
problem Finding minimal surfaces on toroids.
method Equivariant variational methods.
result Linear growth of surface areas with symmetry order.
Constructs minimal surfaces in a 3-ball using PDE gluing.
problem Finding minimal surfaces in a 3-ball with boundary constraints.
method PDE gluing construction of discrete free boundary minimal annuli.
result Discrete family of non-rotational free boundary minimal annuli in a unit 3-ball.
Researchers prove rigidity of first conformal Steklov eigenvalue on specific shapes.
problem Rigidity of the first conformal Steklov eigenvalue on annuli and Möbius bands.
method Proof relies on uniqueness results, compactness theorem, and asymptotic control of Steklov eigenvalues.
result Rigidity of the first conformal Steklov eigenvalue on annuli and Möbius bands proved.
Study on hard Legendrian unknots using normal rulings.
problem Understanding the complexity of Legendrian unknots in knot theory.
method Using normal rulings to obstruct and construct hard unknot diagrams.
result Construction of infinitely many smoothly hard max-tb unknot diagrams with bounds on minimum possible writhe.
We construct two one-parameter families of minimal properly embedded surfaces in the Lie group Sol3 using a Weierstrass-type representation. These surfaces are not invariant by a one-parameter group of ambient isometries. The first one can be viewed as a family of helicoids, and the second one is a family of minimal an…
We analyze how a family of essential annuli in a compact 3-manifold will induce, from a strongly irreducible generalized Heegaard splitting of the ambient manifold, generalized Heegaard splittings of the complementary components. There are specific applications to the subadditivity of tunnel number of knots, improving …
Paper shows how to embed Möbius bands with many twists and small aspect ratios.
problem Finding the smallest aspect ratio for Möbius bands with many twists.
method Constructs a folded paper ribbon knot to bound the aspect ratio.
result Paper Möbius bands and annuli with any number of half-twists can be embedded with aspect ratio less than 8.
Agent finds unknotting sequences for complex knots.
problem Finding minimal crossing changes to unknot complex knots.
method Reinforcement learning agent for knot diagrams.
result Upper bounds on unknotting numbers for thousands of knots.
The (ordinary) unknotting-number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. It is very natural to consider the `unknotting-number' associated with other local-moves on n-dimensional knots, where n is a natural number. In this paper we prove the follow…
The paper examines how the first Steklov-Dirichlet eigenvalue changes with the distance between two concentric circles.
problem Investigating the monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli.
method The approach involves showing differentiability, deriving integral expressions for the derivative, and using variational formulations to find upper and lower bounds.
result The paper proves the monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli with respect to the distance between the centers of the inner and outer boundaries.
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.
New invariant measures how many twists are needed to unknot welded knots.
problem Measuring complexity of welded knots.
method Local twist move, Alexander quandle coloring, Gordian distance.
result Established lower bound on twist number and related it to other knot invariants.