Study uses braid conjugacy class invariants to find ribbon obstructions for knots.
problem Finding effective obstructions for slice knots to be ribbon.
method Apply Rudolph's work and Khovanov-Lee theory to braid conjugacy class invariants.
result Cannot obtain effective ribbon obstructions using Khovanov-Lee theory.
Study on symmetric braid index of ribbon knots, deriving bounds and characterizations.
problem Understanding the symmetric braid index of ribbon knots.
method Defining symmetric braid index, using Khovanov homology, and calculating bounds.
result Existence of knots with symmetric braid index greater than braid index.
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. 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.
A new Markov theorem for 4D ribbon torus-links.
problem Describing isotopic links in R4. method Develops a theorem for ribbon torus-links in B3imesS1. result First step towards a 4D Markov theorem.
Paper proves Markov's theorem for extended welded braids and links.
problem Generalizing welded links to extended welded links and braids.
method Following Kamada's approach, proving Alexander and Markov's theorems for extended welded braids and links.
result Proves versions of Alexander and Markov's theorems for extended welded braids and links.
We study 3-braid knots of finite smooth concordance order. A corollary of our main result is that a chiral 3-braid knot of finite concordance order is ribbon.
Legendrian ribbons and strongly quasipositive links in open books.
problem Understanding links in open books and their properties.
method Using Legendrian ribbons and strongly quasipositive braids.
result Links in open books can be realized as strongly quasipositive braids if they bound Legendrian ribbons.
New categories from TQFTs interpret skein relations.
problem Interpreting skein relations in TQFTs.
method Constructing half-braided algebras and their bimodules.
result Stated skein relations correspond to a TQFT.
Determines surgeries on chain links bounding rational homology balls using lattice-theoretic methods.
problem Integral surgeries on chain links bounding rational homology balls.
method Lattice-theoretic cubiquity obstruction and practical computation methods.
result Proves slice-ribbon conjecture for quasi-alternating 3-braid links, extending previous results.
New Lissajous-toric knots studied with braid representations.
problem Characterizing and understanding Lissajous-toric knots.
method Investigates braid representations and properties of Lissajous-toric knots.
result Upper bounds for 4-genus and examples of trivial knots.
Quantum invariant derived from ternary cohomology of self-distributive structures.
problem Defining and proving a quantum invariant from ternary cohomology.
method Constructing a ribbon category from a TSD set, showing it coincides with the cocycle invariant.
result The ribbon cocycle invariant is a quantum invariant.
We show that every trivial 3-strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result extends to every Artin--Tits group of dihedral type, but it fails to extend to braids with 4 strands and more. The proof uses a partition of the Cayley graph …
Ribbon tangles are proper embeddings of tori and cylinders in the 4-ball~B4, "bounding" 3-manifolds with only ribbon disks as singularities. We construct an Alexander invariant A of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group G. Th…
Functor connects 4D 2-handlebodies to ribbon categories, detecting non-deformation diffeomorphisms.
problem Detecting non-deformation diffeomorphisms in 4D 2-handlebodies.
method Constructs a braided monoidal functor from 4D 2-handlebodies to unimodular ribbon categories.
result Functor J4 detects non-deformation diffeomorphisms when H∗ is not semisimple and H is not factorizable. 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…
We define the notion of a braided link cobordism in S3×[0,1], which generalizes Viro's closed surface braids in R4. We prove that any properly embedded oriented surface W⊂S3×[0,1] is isotopic to a surface in this special position, and that the isotopy can be taken rel boundary wh…
We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group Γ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\…
New braided Frobenius algebras created from specific Hopf algebras.
problem Creating new algebraic structures from Hopf algebras.
method Heap operation and Yang-Baxter operator on tensor product.
result Heap operation induces a braiding compatible with Frobenius operations.
Refines a tangle invariant using XC-algebras.
problem Building a refined tangle invariant using XC-algebras.
method Constructing a canonical strict monoidal functor that refines the Kerler-Kauffman-Radford invariant.
result Preserves the braiding, twist, and open trace.
A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of…
Calculates Dehn twist actions on conformal blocks for modular categories.
problem Order of Dehn twists on spaces of conformal blocks.
method Quantum representations of mapping class groups, ribbon twists, monoidal powers.
result Order of Dehn twists generalizes previous results for small quantum group.
The paper develops a new theory for quantum link invariants using quandles and biquandles.
problem Quantum link invariants for links with SL2(C) flat connections. method Using quandles and biquandles, the paper extends Reshetikhin-Turaev functor to tangles.
result A new invariant of links with a gauge class of quandle representations.
This paper identifies braided 3-belts that can be written in a braid-only form.
problem Identifying braided 3-belts that can be written in a braid-only form.
method Developed an algorithm to calculate the braid word for braided 3-belts and determined the conditions for knotted boundaries.
result Identified the set of braided 3-belts that can be written in a braid-only form and derived a formula for the Jones polynomial for knotted boundaries.
We show that simple coverings of B^4 branched over ribbon surfaces up to certain local ribbon moves bijectively represent orientable 4-dimensional 2-handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over…
New skein categories for non-semisimple settings, extending existing theory.
problem Extending skein theory to non-semisimple settings.
method Introducing skein categories based on tensor ideals in linear ribbon categories.
result Skein categories coincide with factorization homology in non-semisimple settings.
A modular object in a symmetric monoidal bicategory is a Frobenius algebra object whose product and coproduct are biadjoint, equipped with a braided structure and a compatible twist, satisfying rigidity, ribbon, pivotality, and modularity conditions. We prove that the oriented 3-dimensional bordism bicategory of 1-, 2-…
The study connects twist positivity to L-space knots and concordance.
problem Understanding the relationship between twist positivity and knot concordance.
method Analyzing Alexander polynomials and braid indices of knots.
result For twist positive L-space knots, the braid index equals the bridge index.
Study of quasimorphisms and bounded cohomology in braided Thompson groups.
problem Investigate quasimorphisms and bounded cohomology in braided versions of Thompson groups.
method Analyze quasimorphisms and bounded cohomology of various braided Thompson groups.
result Found infinite-dimensional spaces of quasimorphisms in some braided Thompson groups and trivial second bounded cohomology in others.
We introduce a "minimal" Kontsevich integral that generates the original Kontsevich integral while at the same time producing ribbons whose boundaries are the braids on which the minimal Kontsevich integral is evaluated. We generalize the definition of the Kontsevich integral to that of graphs in R^3 and study the beha…
We introduce a new construction of a surface link in the 4-space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering T2-link is equivalent to the split union of spun T2-links and turned spun T2-links. We show th…
We define a category vT of tangles diagrams drawn on surfaces with boundaries. On the one hand we show that there is a natural functor from the category of virtual tangles to vT which induces an equivalence of categories. On the other hand, we show that vT is universal among ribbon c…
We construct what we call a Kirby category, a monoidal category whose morphisms are smooth 4-manifolds, projecting down to another monoidal category whose morphisms are orientable 3-manifolds, the projection being induced by the boundary map on manifolds. We construct a higher categorical generalization of such concept…
This paper defines ribbons and ribbon complexes in CW spaces and analyzes their topological properties.
problem Characterizing and analyzing topological structures in CW spaces.
method Introducing planar ribbons, ribbon complexes, and ribbon nerves in Alexandroff-Hopf-Whitehead CW spaces, and studying their topological properties.
result Characterization of ribbons and ribbon nerves by Betti numbers and homotopy types.
Formula for Alexander polynomials of simple-ribbon knots derived.
problem Determining Alexander polynomials for simple-ribbon knots.
method Introduced simple-ribbon fusions and used them to derive a formula for Alexander polynomials.
result Formula for Alexander polynomials of simple-ribbon knots derived.
Categorifies a skein relation for links colored by one-column Young diagrams.
problem Categorification of a skein relation for links.
method Homotopy equivalence between two complexes of singular Soergel bimodules.
result Categorification of the colored skein relation.
Generalizes ribbonness result for surface-links.
problem Characterizing ribbon surface-links.
method Analyzes handle-irreducible summands and uses equivalences.
result Every stable-ribbon surface-link is a ribbon surface-link.
We generalize bridge trisections to surfaces in four-manifolds, linking them to braided links.
problem Embedding surfaces in four-manifolds and understanding their braiding.
method Introducing bridge trisections, isotoping surfaces, and using trisections and open-book decompositions.
result Any neatly embedded surface can be isotoped to lie in bridge trisection position with respect to any trisection of the four-manifold.
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…
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.
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.
Extends Heisenberg homology to ribbon graphs.
problem Configurations in bounded surfaces.
method Regular thickening of ribbon graphs.
result Heisenberg homology applied to ribbon graphs.
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 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.
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.