arXiv research

Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper we consider ribbon tubes and ribbon torus-link…

This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.

This note investigates the so-called Tube map which connects welded knots, that is a quotient of the virtual knot theory, to ribbon torus-knots, that is a restricted notion of fillable knotted tori in the 4-sphere. It emphasizes the fact that ribbon torus-knots with a given filling are in one-to-one correspondence with…

Extends Satoh's map to virtual m-links and constructs geometric pictures.

The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.

New theory classifies knotted spheres in 4D space.

The paper explores meridional ranks of knotted surfaces and welded knots, proving equalities and relationships.

In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strin…

Weyl's tube formula holds for various cross-sections under symmetry conditions.

Computes tube formulas for valuations in complex space forms.

We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that Satoh's Tube map (from welded links to ribbon knotted tori in $S^4$) is functoria…

The paper defines marginal tubes and proves their null nature.

This article introduces planar ribbons, Vergili ribbon complexes and ribbon nerves in Alexandroff-Hopf-Whitehead CW (Closure finite Weak) topological spaces. A {\em planar ribbon} (briefly, {ribbon}) in a CW space is the closure of a pair of nesting, non-concentric filled cycles that includes the boundary but does not …

In a previous paper, we introduced special types of fusions, so called simple-ribbon fusions on links. A knot obtained from the trivial knot by a finite sequence of simple-ribbon fusions is called a simple-ribbon knot. Every ribbon knot with <10 crossings is a simple-ribbon knot. In this paper, we give a formula for th…

Study proves certain Zoll manifolds with entire Grauert tubes are isometric.

Zero entropy found in entire Grauert tubes of certain manifolds.

Partial answer to affineness of entire Grauert tubes, with Stein manifold criterion.

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…

Lower bound on boundary injectivity radius for specific tubes.

New bounds found for ribbon numbers of knots and links.

H. Hotelling proved that in the n-dimensional Euclidean or spherical space, the volume of a tube of small radius about a curve depends only on the length of the curve and the radius. A. Gray and L. Vanhecke extended Hotelling's theorem to rank one symmetric spaces computing the volumes of the tubes explicitly in these …

New method for flexible tubes and structures, enabling rigid-foldability.

Zoll manifolds with entire Grauert tubes are proven to be standard complex projective spaces.

Study on folded ribbon knots and their minimum length.

We study the ribbon discs that arise from a symmetric union presentation of a ribbon knot. A natural notion of symmetric ribbon number is introduced and compared with the classical ribbon number. We show that the gap between these numbers can be arbitrarily large by constructing an infinite family of ribbon knots with …

In this paper, we analyze the Bollobás and Riordan polynomial $\mathcal{R}$ for ribbon graphs with half-ribbons introduced in [Combinatorics, Probability and Computing 31, 507-549, 2022]. We prove the universality property of a multivariate version of $\mathcal{R}$ whereas $\mathcal{R}$ itself turns out to be universal…

Author provides an alternate proof of the free ribbon lemma.

Extends Heisenberg homology to ribbon graphs.

New knots bound multiple non-isotopic ribbon disks.

The paper calculates ribbon numbers for 12-crossing knots using Alexander polynomials.

Study Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.

Study knots that divide ribbon knotted surfaces, computing their half ribbon genus and fusion number.

Ribbon cobordism forms a partial order in 3-manifolds.

New concept of quasi-ribbon surface-links simplifies complex surface-links.

Paper studies metric ribbon graphs and provides a recursion for their volumes.

Study shows not all ribbon knots can be symmetric unions.

Solves Skopenkov's problem on graph embedding criteria.

We prove recognition theorems for codimension one manifold factors of dimension $n \geq 4$. In particular, we formalize topographical methods and introduce three ribbons properties: the crinkled ribbons property, the twisted crinkled ribbons property, and the fuzzy ribbons property. We show that $X \times \mathbb{R}$ i…

We develop the concept of Cartan ribbons together with a rolling-based method to ribbonize and approximate any given surface in space by intrinsically flat ribbons. The rolling requires that the geodesic curvature along the contact curve on the surface agrees with the geodesic curvature of the corresponding Cartan deve…

Classifies fibered ribbon pretzels, except for a few cases.

Study on volume of tubes and concentration in Riemannian geometry.

This paper investigates symmetric ribbon numbers of low-complexity knots.

Free surface-links are shown to be ribbon links.

Revised proof shows ribbonness of surface-links in 4-sphere.

A surface is called a tube if its level-sets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the so-called flow vector. We study how the geometry of the Gaussian image of a higher-dimensional minimal tube M is controlled by the angle alph…

We study the relationship between fibered ribbon 1-knots and fibered ribbon 2-knots by studying fibered slice disks with handlebody fibers. We give a characterization of fibered homotopy-ribbon disks and give analogues of the Stallings twist for fibered disks and 2-knots. As an application, we produce infinite families…

We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains.The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry …

The paper defines cyclic sets from ribbon string links and connects them to quantum invariants.