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.
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48 results for “ribbon torus-links”
A new Markov theorem for 4D ribbon torus-links.
problem Describing isotopic links in .
method Develops a theorem for ribbon torus-links in .
result First step towards a 4D Markov theorem.
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…
Study on the ribbonlength of knots and links, improving upper bounds.
problem Finding the infimal folded ribbonlength of knots and links.
method Creating new folded ribbon knots and applying them to improve upper bounds.
result Improved upper bounds for -torus links, twist knots, and pretzel links.
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…
We characterize the first Alexander Z[Z]-modules of ribbon surface-links in the 4-sphere fixing the number of components and the total genus, and then the first Alexander Z[Z]-modules of surface-links in the 4-sphere fixing the number of components. Using the result of ribbon torus-links, we also characterize the first…
The paper classifies twisted torus links that are unlinks.
problem Identifying conditions for twisted torus links to be unlinks.
method Characterization of parameter families for unlinking twisted torus links.
result Complete characterization of parameter families for unlinking twisted torus links.
The study finds hyperbolic twisted torus links for certain twists.
problem Identifying hyperbolic twisted torus links.
method Analyzing -torus links with strands twisted times.
result All hyperbolic twisted torus links are identified for .
New method calculates number of components in twisted torus links.
problem Determining the number of components of twisted torus links.
method Euclidean algorithm-like procedure based on parameters.
result Number of components is a multiple of gcd(p, q, r, s).
Torus link homologymath.GT
Study homology of torus links and colored knots.
problem Computing homology of specific link types.
method Khovanov-Rozansky homology for positive torus links and colored knots.
result Computed homology for a family of links including torus links and colored knots.
New findings on T-links derived from torus links.
problem Difficulty in determining geometric type of T-links.
method Examined T-links obtained by full twists along torus links.
result T-links obtained by full twists are not torus links.
Study signatures of torus links and their cores using Neumann's equivariant signatures and Hirzebruch's formula.
problem Computing signatures of torus links and their cores.
method Use Neumann's equivariant signatures and rewrite Hirzebruch's formula for torus links (without cores) in terms of integral points in a parallelogram.
result Rewritten Hirzebruch's formula for torus links with cores using integral points in a parallelogram.
Study reveals action on link homology of specific torus links.
problem Determining action on link homology.
method Using equivariant Khovanov-Rozansky homology framework.
result Identified module structure on link homology.
We introduce a new method for computing triply graded link homology, which is particularly well-adapted to torus links. Our main application is to the (n,n)-torus links, for which we give an exact answer for all n. In several cases, our computations verify conjectures of Gorsky et al relating homology of torus links wi…
Study the geometry of torus link character varieties, finding unexpected relations.
problem Understanding the geometry of torus link character varieties.
method Developed an intrinsic stratification to relate the geometry with torus knots, computed E-polynomial for SL2(C) and SL3(C).
result Unexpected relation with the number of strands of the link.
Characters from logarithmic VOAs linked to torus link invariants.
problem Understanding characters of logarithmic vertex operator algebras.
method Relating characters to coloured Jones invariants of torus links.
result Characters of logarithmic VOAs are limits of coloured Jones invariants of torus links.
Researchers determine quantum filtration structure of torus links.
problem Quantum filtration structure of torus links.
method Determined quantum filtration structure of Lee homology of torus links.
result Confirmed conjectures about torus links and s-invariant.
Formula for twisted Alexander polynomials of torus links.
problem Computing twisted Alexander polynomials for torus links.
method Explicit formula derivation and character variety analysis.
result Locally constant function on character variety.
Study finds minimal coloring numbers for torus links using rack colorings.
problem Determining the minimal number of colors for torus link diagrams.
method Using rack colorings on link diagrams to classify minimal colorings.
result Complete classifications of -colorings by four colors.
The paper introduces an algebraic structure for the stable Khovanov homology of torus links.
problem Studying the stable limit of negative torus links in reduced Khovanov homology.
method Endowing the stable space with a bi-graded commutative algebra structure and describing these algebras explicitly.
result Explicit description of algebras for specific cases and a lower bound for homology width.
Computes knot groups for torus links using Seifert-van Kampen theorem.
problem Computing knot groups for torus links.
method Groupoid version of the Seifert-van Kampen theorem.
result Economical presentations of knot groups for torus links.
The elliptic Hall algebra governs torus link homology.
problem Proving the elliptic Hall algebra's role in torus link homology.
method Developed a rational Shareshian-Wachs involution to prove the symmetry of generating functions.
result Resolved a conjecture by establishing the elliptic Hall algebra's role in torus link homology.
The structure of the Khovanov homology of torus links has been extensively studied. In particular, Marko Stosic proved that the homology groups stabilize as . We show that the Khovanov homotopy types of torus links, as constructed by Robert Lipshitz and Sucharit Sarkar, also become s…
Torus links and -log VOAmath.QA
Study reveals connection between torus links and logarithmic VOAs.
problem Understanding the relationship between torus links and logarithmic VOAs.
method Proposed a geometric method to compute the singlet character of -log VOA.
result The singlet character of -log VOA at the root of unity coincides with the Kashaev invariant and exhibits quantum modularity.
Symmetric function lifts torus link homology.
problem Computing the triply-graded Khovanov-Rozansky homology of torus links.
method Defined a symmetric function and showed it satisfies a recursion for torus link homology.
result Triply-graded Khovanov-Rozansky homology of torus links is a specialization of .
The study characterizes torus links' coloring quivers using dihedral quandles.
problem Characterizing the structure of coloring quivers for torus links.
method Exhaustively determining all possible numbers of colorings and their interconnections.
result The quiver structure varies based on the number of colorings.
Efficient cobordisms show minimal signature values on certain links.
problem Finding minimal signature values for specific link types.
method Constructing topological cobordisms between torus links and connected sums of trefoil knots.
result The signature invariant at takes minimal values on torus links.
Alternative proof of Dynnikov's three-page index for torus links.
problem Proving the existence of a three-page presentation for any link.
method Provided an alternative proof and defined the three-page index.
result Determined exact three-page indices for several torus links.
We compute bordered Floer homology CFDD of (2,2n)-torus link complement, and discuss assorted examples and type-DD structure homotopy equivalence.
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.
A spectral sequence converging to Khovanov homology is constructed which is applied to calculate the rational Khovanov homology of (3,q)-torus links.
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.
This article is about a natural distance function induced by smooth cobordisms between links. We show that the cobordism distance of torus links is determined by the profiles of their signature functions, up to a constant factor.
New minimal link diagrams found, including torus links and homogeneous ones.
problem Finding minimal link diagrams with new classes.
method Morton-Franks-Williams inequality approach.
result New classes of minimal link diagrams, including previously unproven ones.
We show that the Kauffman bracket skein modules of certain manifolds obtained from integral surgery on a (2,2b) torus link are finitely generated, and list the generators for select examples.
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.
We show that the triply graded Khovanov-Rozansky homology of the torus link stablizes as . We explicitly compute the stable homology (as a ring), which proves a conjecture of Gorsky-Oblomkov-Rasmussen-Shende. To accomplish this, we construct complexes of Soergel bimodules which categorify t…
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.
Classifies Legendrian torus and cable links, revealing symmetries and invariants.
problem Classifying Legendrian torus and cable links.
method Classification through Legendrian and transversal isotopies, symmetries analysis.
result First family of non-destabilizable links without maximal Thurston-Bennequin invariant.
Symmetric ribbon disksmath.GT
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 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 whereas 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.
We derive formulas for HOMFLY polynomials of torus links using braid groups and linear recurrences.
problem Calculating HOMFLY polynomials for torus links.
method Using braid groups and linear recurrences, derived from the skein relation.
result Explicit formulas for HOMFLY polynomials of torus links and are derived.
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.