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…
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48 results for “ribbon surfaces”
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.
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.
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.
We introduce the notion of a ribbon-clasp surface-link, which is a generalization of a ribbon surface-link. We generalize the notion of a normal form on embedded surface-links to the case of immersed surface-links and prove that any (immersed) surface-link can be described in a normal form. It is known that an embedded…
Alternative proof for ribbon surfaces in 3D space.
problem Proving ribbonness of surfaces in 3D space.
method Using a compact oriented proper surface in upper half 4-space.
result Link bounds a ribbon surface in upper half 4-space.
Ribbonness on classical linkmath.GT
Ribbonness proven for slice knots, solving an old question.
problem Proving ribbonness for slice knots.
method Showing a link bounds a ribbon surface that is a renewal embedding of a bounding surface.
result Every slice knot is a ribbon knot.
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.
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.
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…
There is a question asking whether a handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link. This question for the case of a trivial surface-link is affirmatively answered. That is, a handle-irreducible summand of every stably trivial surface-link is only a trivial 2-link. By com…
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.
Extends Heisenberg homology to ribbon graphs.
problem Configurations in bounded surfaces.
method Regular thickening of ribbon graphs.
result Heisenberg homology applied to ribbon graphs.
This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.
problem Finite type invariants for ribbon knotted surfaces and their relation to welded string links.
method Developed a theory of finite type invariants for welded string links up to -equivalence, studied algebraic structures, and showed characterizations.
result Characterizes the information contained by finite type invariants in low degrees for welded string links.
We give necessary conditions of a surface-knot to be ribbon concordant to another, by introducing a new variant of the cocycle invariant of surface-knots in addition to using the invariant already known. We demonstrate that twist-spins of some torus knots are not ribbon concordant to their orientation reversed images.
In this paper we study infinitesimal and finite flexibility for generic semidiscrete surfaces. We prove that generic 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of exi…
A quandle coloring obstruction prevents a specific link from being ribbon concordant.
problem Obstructing a specific link from being ribbon concordant.
method Symmetric dihedral quandle coloring analysis.
result A symmetric dihedral quandle of order 4 cannot color a specific surface-link, obstructing ribbon concordance.
Introduces fundamental heaps for surfaces, linking them to cocycle invariants.
problem Isotopy invariants of surface embeddings in 3-space.
method Definition of fundamental heaps using surface ribbons and heap operations.
result Fundamental heaps have a free part whose rank matches the number of connected components.
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.
Study on knot classification using 3-braid closures and ribbon surfaces.
problem Classifying smoothly slice knots from 3-braid closures.
method Construct ribbon surfaces and use twisted Alexander polynomial.
result Classification of knots up to 20 crossings.
Algorithm converts curves on ribbon surfaces to contact surgery diagrams.
problem Legendrian realization of curves on ribbon surfaces.
method Explicit algorithm to Legendrian realize homologically nontrivial curves.
result Any two Legendrian realizations of the same curve are Legendrian isotopic.
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 introduce Khovanov homology for ribbon graphs and show that the Khovanov homology of a certain ribbon graph embedded on the Turaev surface of a link is isomorphic to the Khovanov homology of the link (after a grading shift). We also present a spanning quasi-tree model for the Khovanov homology of a ribbon graph.
Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.
problem Distribution of shapes of complementary subsurfaces in moduli space.
method Study of shapes of complementary subsurfaces in moduli space as boundary lengths go to infinity.
result Random subsurfaces look like random ribbon graphs.
In this paper, we consider local moves on classical and welded diagrams of string links, and the notion of welded extension of a classical move. Such extensions being non-unique in general, the idea is to find a topological criterion which could isolate one extension from the others. To that end, we turn to the relatio…
Helical ribbons arise in many biological and engineered systems, often driven by anisotropic surface stress, residual strain, and geometric or elastic mismatch between layers of a laminated composite. A full mathematical analysis is developed to analytically predict the equilibrium deformed helical shape of an initiall…
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram , there is an associated ribbon graph whose quasi-trees correspond bijectively to …
Quantum invariants for surfaces in 4D 2-handlebodies.
problem Quantum invariants of ribbon surfaces in 4D 2-handlebodies.
method Unimodular ribbon categories, labeled Kirby graphs, and modified traces.
result Recovery and generalization of existing invariants.
We construct an orientable ribbon surface F in B^4, which is universal in the following sense: any compact orientable pl 4-manifold having a handle decomposition with 0-, 1- and 2-handles can be represented as a cover of B^4 branched over F.
We generalize the idea of unknotting knots to Seifert surfaces. We define an operation called ribbon twist which serves as the equivalent of a crossing change for knots. A Seifert surface is considered untwisted, the equivalent to unknotted, if it is isotopic to a standardly embedded n-fold punctured torus. A Seifert s…
We discuss how to apply work of L. Rudolph to braid conjugacy class invariants to obtain potentially effective obstructions to a slice knot being ribbon. We then apply these ideas to a family of braid conjugacy class invariants coming from Khovanov-Lee theory and explain why we do not obtain effective ribbon obstructio…
In this paper we exhibit infinite families of embedded tori in 4-manifolds that are topologically isotopic but smoothly distinct. The interesting thing about these tori is that they are topologically trivial in the sense that each bounds a topologically embedded solid handlebody. This implies that there are stably ribb…
In the present work we are going to give a formal exposition of the ribbon graphs topic based on notes of Labourie \cite{Lab}, since is difficult to find as such in the literature.
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…
New invariant shows fifth move is unique and extends biquandle theory for surface-links.
problem Reproduce Yoshikawa's fifth move using other moves and isotopies.
method Develops a semi-invariant and biquandle-based invariant for surface-links.
result Yoshikawa's fifth move is unique and cannot be reproduced by other moves.
This paper extends results of Hatcher and Vogtmann's work "Cerf Theory for Graphs" to ribbon graphs. Given an orientable, punctured and basepointed surface Sigma, we prove that the space of ribbon graphs that can be drawn in Sigma is filtered by simplicial complexes. The k-th simplicial complex is (k-1)-dimensional, (k…
We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies $\sel(K,S)=-χ(S)$. In particular, every null-homolog…
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollobás-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chor…
Python tool calculates cobordism maps in Khovanov homology.
problem Computing cobordism maps on Khovanov homology.
method Developed a Python module to calculate these maps.
result Computed cobordism maps for all incompressible Seifert surfaces of prime knots up to 10 crossings.
Study fundamental quandle of ribbon concordances, proving homomorphisms.
problem Understanding fundamental quandle of ribbon concordances.
method Topological definition of fundamental quandle, motion picture diagrams.
result Injective and surjective quandle homomorphisms from ribbon concordance.
In this note, I discuss in some detail the dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces with boundary and marked points, which I introduced in math.AG/0402015, and used in math.QA/0412149 to construct open-closed topological conformal field theories. This dual version of the r…
We consider knotted annuli in 4-space, called 2-string-links, which are knotted surfaces in codimension two that are naturally related, via closure operations, to both 2-links and 2-torus links. We classify 2-string-links up to link-homotopy by means of a 4-dimensional version of Milnor invariants. The key to our proof…
For every n-component ribbon link L we prove that the Jones polynomial V(L) is divisible by the polynomial V(O^n) of the trivial link. This integrality property allows us to define a generalized determinant det V(L) := [V(L)/V(O^n)]_(t=-1), for which we derive congruences reminiscent of the Arf invariant: every ribbon …
Involutive Hopf monoids yield surface invariants.
problem Involutive Hopf monoids in symmetric monoidal categories.
method Construction of invariants via (co)equalizers and images.
result Categorical generalization of quantum double models.
This paper characterizes extensions of augmented racks and constructs invariants for surfaces.
problem Characterizing extensions of augmented racks and constructing invariants for surfaces.
method Characterization of rack extensions through fibrant and additive cohomology, construction of invariants using cocycles.
result Characterization of extensions of augmented racks and construction of surface invariants.
Extends Satoh's map to virtual m-links and constructs geometric pictures.
problem Virtual m-links and their geometric representations.
method Generalization of Satoh's map to virtual m-links and construction of geometric pictures.
result New geometric pictures of virtual m-links.
A criterion is given for cutting out disks with ribbons from a Möbius strip.
problem Determining which hieroglyphs can be realized as disks with ribbons on a Möbius strip.
method Developed a criterion based on Mohar's realizability criterion, leading to a quadratic algorithm.
result A criterion for weak realizability of disks with ribbons on a Möbius strip.
We give an example of a 3-component smoothly slice boundary link, each of whose components has a genus one Seifert surface, such that any metaboliser of the boundary link Seifert form is represented by 3 curves on the Seifert surfaces that form a link with nonvanishing Milnor triple linking number. We also give a gener…