Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.
problem Understanding the structure of 2-complexes and their asphericity.
method Using Kervaire's sphere-link and ribbon sphere-link equivalence, analyzing the compact complement of ribbon disk-links.
result Every connected subcomplex of a contractible finite 2-complex is aspherical.
Author provides an alternate proof of the free ribbon lemma.
problem Proving that every free sphere-link in the 4-sphere is a ribbon sphere-link.
method An alternate proof of the free ribbon lemma.
result Provides an alternate proof of the free ribbon lemma.
Free surface-links are shown to be ribbon links.
problem Characterizing surface-links as ribbon links.
method Four proofs are provided, including a stabilization approach.
result Every free surface-link is a ribbon surface-link.
Paper confirms Whitehead's conjecture for aspherical 2-complexes.
problem Whitehead's conjecture about aspherical 2-complexes.
method Argument on ribbon sphere-links, generalized for aspherical 2-complexes.
result Whitehead's conjecture confirmed for aspherical 2-complexes.
This note proves properties of surface-links with trivial components.
problem Properties of surface-links with trivial components.
method Analyzes closed oriented disconnected surfaces and their links.
result For certain surface-links, trivial components result in ribbon links.
Perelman's proof confirmed, new method uses 4D topology.
problem Confirming the classical Poincaré conjecture.
method 4D topology, spun torus-knots, ribbonness, disk-chord system, Bing's result.
result Homotopy 3-sphere is diffeomorphic to the 3-sphere.
It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…
While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities…
In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to co…
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.
problem Finding bounds for the ribbon number of knots and links.
method Using determinant and Jones polynomial, we find new lower bounds and prove the finiteness of certain sets.
result We determine the set of Jones polynomials for ribbon knots with 11 or fewer crossings.
Study on folded ribbon knots and their minimum length.
problem Finding the minimum length of folded ribbon knots.
method Using Kauffman's model of folded ribbon knots and analyzing their properties.
result Proved bounds on the minimum folded ribbonlength for various types of knots.
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 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 R whereas R itself turns out to be universal…
Extends Heisenberg homology to ribbon graphs.
problem Configurations in bounded surfaces.
method Regular thickening of ribbon graphs.
result Heisenberg homology applied to ribbon graphs.
New knots bound multiple non-isotopic ribbon disks.
problem Finding knots that bound multiple non-isotopic ribbon disks.
method Classification of fibered, homotopy-ribbon disks for generalized square knots.
result Infinitely many knots bound infinitely many pairwise non-isotopic ribbon disks.
The paper calculates ribbon numbers for 12-crossing knots using Alexander polynomials.
problem Determining the minimum number of ribbon singularities for knots.
method Using Alexander polynomials and systematic treatment of knot invariants.
result Computed ribbon numbers for many 12-crossing knots.
Study Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.
problem Determining Alexander polynomials for ribbon and virtual knots.
method Using ribbon's intrinsic singularity information, defining half Alexander polynomial, and developing simplified formulas.
result New formulas for Alexander polynomials of general knots and virtual knots in terms of Gauss diagrams.
Study knots that divide ribbon knotted surfaces, computing their half ribbon genus and fusion number.
problem Understanding knots that divide ribbon knotted surfaces and their properties.
method Defining half ribbon knots, computing half ribbon genus and fusion number, and comparing with Levine-Tristram signatures.
result Computed half ribbon genus and fusion number for various knots, including new computations of doubly slice genus.
Ribbon cobordism forms a partial order in 3-manifolds.
problem Understanding partial orders in 3-manifolds.
method Utilizing recent methods from Ian Agol's work on knot concordance.
result Ribbon rational homology cobordism forms a partial order.
Study topological Iwasawa invariants for 3-sphere links, proving density results.
problem Detect and analyze Iwasawa invariants for 3-sphere links.
method Explicit criteria for detecting Iwasawa invariants, statistical analysis of 2-bridge links.
result Density of 2-bridge links with specific Iwasawa invariants.
New concept of quasi-ribbon surface-links simplifies complex surface-links.
problem Complexity in surface-links of trivial components.
method Introducing quasi-ribbon surface-links as a generalization of ribbon surface-links.
result Every F-link of trivial components on a surface F with at most one aspheric component is a quasi-ribbon surface-link.
Paper studies metric ribbon graphs and provides a recursion for their volumes.
problem Calculating volumes of combinatorial moduli spaces of directed metric ribbon graphs.
method Decomposes directed ribbon graphs into simpler graphs with one vertex, proving a canonical recursion scheme for volumes.
result Explicit recursion for volumes of four-valent metric ribbon graphs provided.
Study shows not all ribbon knots can be symmetric unions.
problem Whether every ribbon knot can be a symmetric union.
method Exhibited a specific ribbon Montesinos knot that cannot be a symmetric union.
result Found a ribbon knot that is not a symmetric union.
Solves Skopenkov's problem on graph embedding criteria.
problem Criteria for toroidal embedding of one-vertex ribbon graphs.
method Analyzes one-vertex ribbon graphs with additional disc structure.
result Provides solutions to Skopenkov's problem.
We prove recognition theorems for codimension one manifold factors of dimension n≥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×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.
problem Classifying fibered ribbon pretzel knots up to mutation.
method Combining lattice embedding techniques with Gabai's classification of fibered pretzel knots, and exhibiting ribbon disks.
result Complete classification except for a few cases.
This paper investigates symmetric ribbon numbers of low-complexity knots.
problem Determining the minimum number of ribbon singularities in symmetric ribbon disks for knots with up to 12 crossings.
method Systematic investigation using knot polynomials and determinants.
result Novel lower bounds for symmetric ribbon numbers of knots with up to 12 crossings.
Revised proof shows ribbonness of surface-links in 4-sphere.
problem Determining ribbonness of surface-links in 4-sphere.
method Surgery along fusion 1-handle systems to prove ribbonness.
result Surface-links in 4-sphere are ribbon if certain conditions are met.
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.
problem Defining and relating cyclic sets from ribbon string links.
method Endowing ribbon string links with cyclic and cocyclic structures, relating to coend of a ribbon category via quantum invariants.
result Established a relationship between ribbon string links and 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.
problem Prove strong ribbon concordance induces a partial order on links.
method Use results from knot Floer homology to certify minimality under the ribbon partial order.
result Certify minimality for a handful of knots and find minimal ribbon minimal knots.
Study ribbon homology concordances using link Floer homology.
problem Understanding ribbon homology concordances and their effects on link Floer homology.
method Combining results from Daemi, Lidman, Vela-Vick, Wong, and Zemke, using link Floer homology and torsion submodules.
result Ribbon homology concordances induce split injections on HFL−. 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.
problem Understanding the concordance properties of knots and their cables.
method Using Seifert genus and concordance invariant γ0 from bordered Heegaard Floer homology.
result Connected sums of γ0-sharp fibered knots are ribbon only if they are of a specific form, or the slice-ribbon conjecture fails.
New equivalence relation on ribbon graphs connects to virtual links.
problem Understanding virtual links through ribbon graphs.
method Introducing a new equivalence relation on ribbon graphs.
result Correspondence between virtual links and ribbon graphs.
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.
problem Understanding ribbon concordance and minimal compressions of surface homeomorphisms.
method Proving monotonicity of simplicial volume and dilatation under ribbon concordance, algorithmic enumeration of minimal compressions.
result Every fibered knot has only finitely many predecessors in the ribbon-concordance partial order.
New ribbon disks in 4D space, non-isotopic to each other.
problem Non-isotopic ribbon disks in 4D.
method Using corks to construct diffeomorphic ribbon disks.
result Non-isotopic ribbon disks constructed in 4D.
New theorem for 4D links simplifies characterisation problem.
problem Long-standing open problem in link characterisation.
method Reidemeister Theorem for solid ribbon torus links.
result Complete characterisation of a related class of links.
Knots can be ordered by ribbon concordance, solving a long-standing question.
problem Ordering knots by ribbon concordance.
method Representation varieties of knot groups to SO(N) and relations induced by ribbon concordance. result Ribbon concordance forms a partial ordering on the set of knots.
Adding a braid closure to a fibered knot makes a link ribbon concordance minimal.
problem Finding the minimal ribbon concordance for links in S3. method Link Floer homology combined with classical techniques.
result Any link L in S3 can be made ribbon concordance minimal by adding a single unknot. 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.
problem Characterizing two-bridge ribbon knots.
method Symmetric union presentations and partial knot analysis.
result Symmetric union presentations for various two-bridge ribbon knots.
The study limits the number of ribbon concordant fibered knots.
problem Understanding the relationship between ribbon concordance and fibered knots.
method Combining Floer homology results with fixed points of monodromy and volume inequalities.
result There are only finitely many hyperbolic fibered knots ribbon concordant to any given knot.