arXiv research

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…

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…

New bounds found for ribbon numbers of knots and links.

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.

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.

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…

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

Prove strong ribbon concordance induces a partial order on links, certify minimality for a handful of knots, and find minimal ribbon minimal knots.

Study ribbon homology concordances using link Floer homology.

A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon.

Sharp knots and iterated cables lead to ribbon knots or failure of slice-ribbon conjecture.

New equivalence relation on ribbon graphs connects to virtual links.

In this paper we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus the ribbon graph minor relation is incompatible wi…

Study ribbon concordance and minimal compressions, proving new results about fibered knots.

New ribbon disks in 4D space, non-isotopic to each other.

New theorem for 4D links simplifies characterisation problem.

Knots can be ordered by ribbon concordance, solving a long-standing question.

Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.

Adding a braid closure to a fibered knot makes a link ribbon concordance minimal.

We consider ribbon n-knots for n\geq 2. For such knots we define a set of moves on ribbon disks, and show that any two ribbon disks for isotopic knots are related by a finite sequence of such moves and ambient isotopies. Using this we are able to prove that there is a natural geometric correspondence between ribbon n-k…

Two-bridge ribbon knots have symmetric union presentations.

The study limits the number of ribbon concordant fibered knots.

Determines conditions for ribbon cobordisms between lens spaces.

New insights into knot fusion numbers via cabling.

Ribbon: Scalable Approximation and Robust Uncertainty Quantification

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…

Ribbon cobordisms form a partial order on 3-manifolds.

We consider ribbon $R^4$'s, that is, smooth open 4-manifolds, homeomorphic to $R^4$ and associated to $h$-cobordisms between closed 4-manifolds. We show that any generalized ribbon $R^4$ associated to a sequence of $h$-cobordisms between non-diffeomorphic 4-manifolds is exotic. Notion of a positive ribbon $R^4$ is defi…

Every link diagram can be represented as a signed ribbon graph. However, different link diagrams can be represented by the same ribbon graphs. We determine how checkerboard colourable diagrams of links in real projective space, and virtual link diagrams, that are represented by the same ribbon graphs are related to eac…