arXiv research

New method to construct slice knots using annulus modifications.

The JSJ decomposition helps classify genus two handlebody-knots.

Defines annulus complex of handlebodies and proves its connectivity.

New homology for links in annulus discovered.

Study shows mean action of periodic orbits in annuli is bounded by their Calabi invariant.

Study on spectral stability of an embedded annulus under curve shortening and Ricci flows.

Solves H-system on annulus with Neumann conditions.

Study proves the critical catenoid is the free boundary minimal annulus.

Curve shortening flow increases annulus modulus.

We present an extension of Dunwoody's theory of tracks and use it to prove an analogue of the annulus theorem for hyperbolic groups.

Paper introduces an invariant to distinguish handlebody-knot exteriors.

Given two Jordan curves in a Riemannian manifold, a minimal surface of annulus type bounded by these curves is described as the harmonic extension of a critical point of some functional (the Dirichlet integral) in a certain space of boundary parametrizations. The $H^{2,2}$-regularity of the minimal surface of annulus t…

Study confirms infinitely many non-characterizing slopes for various knots.

We show that a compact embedded minimal or constant mean curvature annulus with non-vanishing Gaussian curvature which is tangent to two spheres of same radius or tangent to a sphere and meeting a plane in constant contact angle is rotational.

The paper extends knotoid theory to annular and toroidal settings.

Unstable minimal surfaces are the unstable stationary points of the Dirichlet-Integral. In order to obtain unstable solutions, the method of the gradient flow together with the minimax-principle is generally used. The application of this method for minimal surfaces in the Euclidean spacce was presented in \cite{s3}. We…

We give a new construction of slice knots via annulus twists. The simplest slice knots obtained by our method are those constructed by Omae. In this paper, we introduce a sufficient condition for given slice knots to be ribbon, and prove that all Omae's knots are ribbon.

We describe the space of arrow diagram formulas for virtual knot diagrams in the annulus as the kernel of a linear map, inspired from a conjecture due to M. Polyak. As a main application, we slightly improve Grishanov-Vassiliev's theorem for planar chain invariants.

We find a self-linking number formula for a given null-homologous transverse link in a contact manifold that is compatible with either an annulus or a pair of pants open book decomposition. It extends Bennequin's self-linking formula for a braid in the standard contact $3$-sphere.

Paper introduces new homology for links in an annulus.

The paper proves the existence of a folded annulus with multiple creases.

Minimal surfaces reflect across spheres, proving annulus uniqueness.

Define web algebras for annular SL(2) and SL(3) using foam TQFTs.

Using a new method we give elementary estimates for the capacity of non-contractible annuli on cylinders and provide examples, where these inequalities are sharp. Here the lower bound depends only on the area of the annulus. In the case of constant curvature this lower bound is obtained with the help of a symmetrizatio…

We show a non existence result for solutions of the prescribed mean curvature equation in the product manifold $\mathbb{H}^2 \times \R$, where $\mathbb{H}^2$ is the real hyperbolic plane. More precisely we prove a-priori estimates for graphs with constant mean curvature $h \in (0, 1/2]$ on circular annuli of $\mathbb{H…

The paper defines conditions for surface sums of handlebodies to be handlebodies.

Using the duality between Wilson loop expectation values of SU(N) Chern-Simons theory on $S^3$ and topological open-string amplitudes on the local mirror of the resolved conifold, we study knots on $S^3$ and their invariants encoded in colored HOMFLY polynomials by means of topological recursion. In the context of the …

New method for knot closures from 1-tangles and annulus twists.

Study on stability of mass theorems using foliated IMCF.

New Gram determinant from Möbius band connects to annulus case.

Generalized Gram determinant for 3-manifold invariants.

The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on an annulus than on any other surface of revolution in $\mathbb{R}^3$ with the same boundary. This is established by defining a sequence of shrinking cylinders about the axis of symmetry and proving that flattening a surface outside of each cylinde…

Study proves uniqueness of certain minimal surfaces in spherical and hyperbolic spaces.

New method characterizes minimal surfaces in 3D space.

Assume that $(\mathcal{N},\hbar)$ and $(\mathcal{M},\wp)$ are two Riemann surfaces with conformal metrics $\hbar$ and $\wp$. We prove that if there is a harmonic homeomorphism between an annulus $\mathcal{A}\subset \mathcal{N}$ with a conformal modulus $\mathrm{Mod}(\mathcal{A})$ and a geodesic annulus $A_\wp(p,ρ_1,ρ_2…

The paper studies cylindrical handlebody-knots of genus two with unique unknotting annuli and finds trivial symmetry groups.

Khovanov homology for links in S^3 via 1-tangle diagrams in annulus.

Categorifies weight-preserving maps in sl(2) representations.

We define composite DAHA-superpolynomials of torus knots, depending on pairs of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the theory of the skein of the annulus. We provide various examples. Our superpolynomials extend the DAHA-Jones (refined) polynomials and satisfy all standard symmetries…

The Hecke algebra H_n contains well known idempotents E_λ which are indexed by Young diagrams with n cells. They were originally described by Gyoja. A skein theoretical description of E_λ was given by Aiston and Morton. The closure of E_λ becomes an element Q_λ of the skein of the annulus. In this skein, they are known…

Khovanov homology remains unchanged by certain surface surgeries.

Characterizes the critical catenoid using nodal domain arguments and Steklov eigenvalue criteria.

This paper defines a functor for $L_\infty$-modules and applies it to Khovanov homology.

A manifold M is simple if it contains no essential disk, sphere, annulus or torus. If M is simple and two Dehn fillings M(r_1), M(r_2) are nonsimple, then there is an upper bound on Δ(r_1,r_2), the geometric intersection number between r_1 and r_2. There are 10 possibilities, depending on the types of M(r_i). In this p…

The meridian maps of the full Homfly skein of the annulus are linear endomorphisms induced by the insertion of a meridian loop, with either orientation, around a diagram in the annulus. The eigenvalues of the meridian maps are known to be distinct, and are indexed by pairs of partitions of integers p and n into k and k…

New categories help understand knot algebra.

We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.

New degenerate free boundary minimal annuli found in spherical caps, challenging uniqueness.