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.
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.
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. 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. 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.
Classifies knotted annuli in 4-space up to a specific equivalence.
problem Classifying knotted annuli in 4-space up to a specific equivalence.
method Uses Milnor invariants and a 4-dimensional version of them, along with a Roseman-type result for immersed surfaces.
result Classifies 2-string-links up to link-homotopy.
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 …
Study on cylindrical handlebody-knots with symmetry and rigidity properties.
problem Characterizing symmetry groups of cylindrical handlebody-knots of genus two.
method Classification of essential annuli and analysis of symmetry groups based on Koda-Ozawa theorem.
result Most exteriors of genus two cylindrical handlebody-knots contain no essential disks or tori, and type 3-3 annuli are often unique up to isotopy. 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.
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.
Paper introduces an invariant to distinguish handlebody-knot exteriors.
problem Challenges in distinguishing handlebody-knots with homeomorphic exteriors.
method Defined an invariant (annulus diagram) using characteristic submanifold theory and Koda-Ozawa classification for essential annuli.
result The annulus diagram can differentiate handlebody-knot families.
This paper studies the question of whether minimal genus Heegaard splittings of exterior spaces of knots which are connected sums are weakly reducible or not. Furthermore it is shown that the Heegaard splittings of the knots used by Morimoto to show that tunnel number can be sub-additive are all strongly irreducible. T…
Study of knots in homology spheres and their concordance properties.
problem When can knots in homology spheres be reduced to slice knots?
method Analyzes concordance and crossing changes in homology spheres.
result Knots in homology spheres are nullhomotopic in a smooth homology ball if and only if they are concordant to a smoothly slice knot.
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.
We study the way a strongly irreducible Heegaard surface Σ intersects a knot exterior X embedded in a 3-manifold, and show that if Σ∩∂X consists of simple closed curves which are essential in both Σ and ∂X, then the intersection X∩Σ consists of meridional annuli only. As an applicat…
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.
2-knots with S4 symmetry are classified up to equivariant concordance.
problem Classifying 2-knots with S4 symmetry up to equivariant concordance. method Constructing a new invariant called periodic, based on the Arf invariant.
result The smooth equivariant concordance group of 2-knots in S4 is isomorphic to Z/2Z for all d≥2. 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.
Study handles in sutured manifolds and knots, finding varied handle numbers and unique surfaces.
problem Understanding handle numbers and incompressible Seifert surfaces in sutured manifolds and nearly fibered knots.
method Extending Haken's Theorem, analyzing product annuli and disks, and examining specific knot types.
result Variety of handle numbers and unique incompressible Seifert surfaces in nearly fibered knots.
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 …
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 : \…
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.
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…
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.
Study of minimal annuli in hyperbolic space with horizontal ends, showing constraints on boundary curves.
problem Constraints on boundary curves of properly embedded minimal annuli in H2imesR. method Analysis of moduli space of properly Alexandrov-embedded, minimal annuli with horizontal ends.
result Boundary curves of minimal annuli are not fully prescribable, but the bottom curve and neck position are fixed, with top curve up to translation and tilt.
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.
Study of knot complements yields quantum modularity insights.
problem Understanding quantum invariants of knot complements.
method Large-N analysis of q-series invariants, counts of holomorphic curves. result Closed-form expressions for a-deformed FK for (2,2p+1)-torus knots. 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…
The paper calculates a formula for knot complements using holomorphic curves.
problem Calculating the partition function of knot complements.
method Skein valued holomorphic curve counting techniques.
result The partition function localizes on specific holomorphic annuli for torus knots.
Study on exotic smooth embeddings of surfaces in 4-manifolds, revealing different properties and complexities.
problem Understanding exotic smooth embeddings of surfaces in 4-manifolds.
method Analyzing smooth, proper embeddings of noncompact surfaces in 4-manifolds, focusing on exotic planes and annuli.
result Exotic planes and annuli exhibit radically different properties, with one class being simple enough to draw explicit level diagrams.
Classifies S1-invariant free boundary minimal annuli and Möbius bands in Bn.
problem Classifying S1-invariant free boundary minimal annuli and Möbius bands in Bn. method Analysis of the spectrum of the Dirichlet-to-Neumann map for S1-invariant metrics. result Existence and classification of S1-invariant free boundary minimal annuli and Möbius bands in Bn. 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.
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.
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 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…
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…
[Original abstract (1992):] The modulus of quasipositivity q(K) of a knot K was introduced as a tool in the knot theory of complex plane curves, and can be applied to Legendrian knot theory in symplectic topology. It has also, however, a straightforward characterization in ordinary knot theory: q(K) is the supremum of …
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.
New analysis of crushing surfaces of positive genus impacts triangulation complexity.
problem Crushing surfaces of positive genus can drastically change triangulation topology.
method Detailed analysis of crushing effects on closed essential surfaces of positive genus.
result Proved results about triangulation complexity and JSJ decompositions.
Let S(D) be the surface produced by applying Seifert's algorithm to the oriented link diagram D. I prove that if D has no negative crossings then S(D) is a quasipositive Seifert surface, that is, S(D) embeds incompressibly on a fiber surface plumbed from positive Hopf annuli. This result, combined with the truth of the…