New mixed singularities help classify real algebraic links.
problem Classify real algebraic links in the 3-sphere.
method Construct new mixed singularities.
result New mixed singularities may help solve the Benedetti-Shiota conjecture.
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48 results for “algebraic links”
Study of algebraic links in lens spaces, proving they are fibered and finding examples.
problem Understanding algebraic links in lens spaces.
method Investigation of polynomials invariant under a -action on the 3-sphere, studying links and their properties.
result All algebraic links in lens spaces are fibered, with results on Seifert genus.
New method uses algebras to speed up link Floer homology calculations.
problem Computing link Floer homology efficiently.
method Using bordered algebras to compute link Floer homology.
result Fast computation of the Thuston polytope for links.
We introduce a two-parameters bt-algebra which, by specialization, becomes the one-parameter bt-algebra, introduced by the authors, as well as another one-parameter presentation of it; the invariant for links and tied links, associated to this two-parameter algebra via Jones recipe, contains as specializations the inva…
New invariant for singular links via bt-algebra.
problem Invariants for singular links.
method Representations of singular braid monoid into two parameter bt-algebra.
result More powerful than previous invariants.
We present a construction of invariants for links using an isomorphism theorem for affine Yokonuma--Hecke algebras. The isomorphism relates affine Yokonuma--Hecke algebras with usual affine Hecke algebras. We use it to construct a large class of Markov traces on affine Yokonuma--Hecke algebras, and in turn, to produce …
New algebra counts components of arborescent knots and links.
problem Counting components of arborescent knots and links.
method Developed a new algebra called the crossing algebra.
result The crossing algebra counts the number of components for arborescent knots and links.
The quadratic linking degreemath.AG
Introduces quadratic linking degree in algebraic geometry.
problem No specific problem stated; focuses on new concept.
method Motivic homotopy theory and Witt group.
result Explicit computation of quadratic linking degree.
We link Ginzburg algebras to Weinstein manifolds and Legendrian knots.
problem Understanding the relationship between Ginzburg algebras and Weinstein manifolds.
method Associated a stopped Weinstein manifold to a quiver and subquiver, proving quasi-isomorphism of relative Ginzburg algebra and Chekanov-Eliashberg dg-algebra.
result Relative Ginzburg algebra is quasi-isomorphic to Chekanov-Eliashberg dg-algebra of a singular Legendrian unknot link.
Witt algebra acts on Khovanov-Rozansky homology of links.
problem Understanding algebraic structures in knot theory.
method Construction of Witt algebra action on Khovanov-Rozansky homology.
result Induced maps between twists of the homology via link cobordisms.
We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0 and a framing relation L^{(1)} = aL, where a, b_0, b_3 are invertible. We introdule the concept of n-algebraic tangles (and links) and analyze the skein module for 3-a…
Delta-unlinking number measures how to unlink algebraically split links.
problem Measuring unlinking complexity of algebraically split links.
method Defining delta-unlinking number as minimum delta-moves to unlink, proving bounds and calculating specific values.
result Precise delta-unlinking numbers for algebraically split prime links up to 9 crossings, and 4-genus values for most.
New methods for delta-moves on algebraically split links identified.
problem Understanding delta-moves on algebraically split links.
method Introducing self and mixed delta-moves, proving equivalence, and calculating delta-splitting numbers.
result Two links are mixed delta-equivalent if they have the same pairwise linking number and components.
Floer homology connects to quiver Hecke algebras in Coulomb branches.
problem Understanding the representation theory of quiver Hecke algebras.
method Using Lagrangian Floer homology in multiplicative Coulomb branches.
result Cylindrical KLRW algebras are realized by Floer homology.
Let be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle . Then separates the strings of in and the boundary slope of is uniquely determined by and hence we can define the slope of the algebraic tang…
Paper extends algebraic geometry results to hyperbolic link complements.
problem Understanding algebraic and number-theoretic properties of canonical curves.
method Generalizes Chinburg-Reid-Stover's results to hyperbolic link complements.
result Azumaya algebra does not extend to canonical surfaces.
This paper discusses the construction of a generalized Alexander polynomial for virtual knots and links, and the reformulation of this invariant as a quantum link invariant. The algebraic background for the generalized Alexander module is formulated in terms of the biquandle, a generalization of the quandle of David Jo…
Study extends knot polynomials to links, identifying them with known invariants.
problem Extending knot polynomials to links.
method Applying Reshetikhin-Turaev functor to braided Hopf algebras with automorphisms.
result Identifies some knot polynomials with known link invariants.
In this paper we announce the existence of a family of new -variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma-Hecke algebra of type . Yokonuma-Hecke algebras are generalizations of Iwahori-Hecke algebras, and this family contains the Homflypt polynomial, the fa…
Paper constructs a HOMFLYPT-type invariant for pseudo links.
problem Inability to construct polynomial invariants for pseudo links using Hecke algebra techniques.
method Using a resolution homomorphism and pseudo Hecke algebra of type \(A\), the paper constructs a HOMFLYPT-type invariant for oriented pseudo links.
result The constructed invariant satisfies a natural pseudo skein relation and admits a state-sum formulation.
New proof found for Khovanov's assertion about Frobenius algebra twists.
problem Proving the isomorphism between chain complexes of twisted Frobenius algebras and link diagrams.
method Detailed analysis of configurations of circles in each state.
result A new proof of Khovanov's assertion with a detailed analysis of configurations.
We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard-Floer link homology of the local embedded link of the germ, (b) the lattice homology asso…
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.
This paper defines the concept of an oriented quantum algebra and develops its application to the construction of quantum link invariants. We show that all known quantum link invariants can be put into this framework.
In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a m…
The paper explains why a specific type of link homology is useful.
problem Link homology theories and their applications.
method Explains the usefulness of gl(N) theories instead of sl(N) for link homology.
result It is useful to consider gl(N) theories for link homology.
In this survey we collect all results regarding the construction of the Framization of the Temperley-Lieb algebra of type as a quotient algebra of the Yokonuma-Hecke algebra of type . More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimensi…
This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up the background virtual knot theory, defines rotational virtual knot theory, stu…
Proof confirms conjecture for certain braids and their closures.
problem Conjecture about real algebraic links and fibered links.
method Analyzes T-homogeneous and related braids, proving conjecture for their closures.
result Conjecture confirmed for closures of T-homogeneous braids.
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 paper connects Apollonian packings to knot theory and improves link representations.
problem Realizing algebraic links in Apollonian packings.
method Introducing new representations of links in tangency graphs of sphere packings, proving link realizability, and improving upper bounds.
result Any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing.
The family of negative torus links over a fixed number of strands admits a stable limit in reduced Khovanov homology as grows to infinity. In this paper, we endow this stable space with a bi-graded commutative algebra structure. We describe these algebras explicitly for . As an application, w…
New algebra structure for Legendrian knots preserves contact homology invariants.
problem Constructing an algebra for Legendrian knots.
method Combining rational Symplectic Field Theory and combinatorial methods.
result Invariant Poisson algebra of Legendrian links under isotopy.
We provide necessary conditions for the Alexander polynomials of algebraically split component-preservingly amphicheiral links. We raise a conjecture that the Alexander polynomial of an algebraically split component-preservingly amphicheiral link with even components is zero. Our necessary conditions and some examples …
Knot lattice homology invariant of smooth knot type in rational homology spheres.
problem Invariance of knot lattice homology in rational homology spheres.
method Proving knot lattice homology invariant through doubly-filtered homotopy type.
result Knot lattice homology invariant of smooth knot type in rational homology spheres.
New algebraic structures help categorify link invariants.
problem Classifying and distinguishing links and virtual links.
method Introducing mc-biquandles and categorifying homsets.
result New link invariants defined via mc-biquandle coloring quivers.
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.
We define a finite-dimensional cubic quotient of the group algebra of the braid group, endowed with a (essentially unique) Markov trace which affords the Links-Grould invariant of knots and links. We investigate several of its properties, and state several conjectures about its structure.
In this paper we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma-Hecke algebras , which are not topologically equivalent to the Homflypt polynomial. We then present the algebra which is the appropriate Temperley-Lieb analogu…
Legendrian solid-torus linksmath.SG
Differential graded algebra invariants are constructed for Legendrian links in the 1-jet space of the circle. In parallel to the theory for R^3, Poincare-Chekanov polynomials and characteristic algebras can be associated to such links. The theory is applied to distinguish various knots, as well as links that are closur…
New algebraic structure for 2-string links and long knots.
problem Understanding the space of 2-string links and long knots.
method Realized as a free algebra over a colored operad .
result Expressed the isotopy class of a 2-string link in terms of its prime factors.
The paper explores connections between braids, links, and cobordisms using algebraic methods.
problem Investigating functions on manifolds and their connections to braids, links, and cobordisms.
method Algebraic methods including group theory, sheaves, and formal groups.
result Constructs Lazard's one-dimensional universal commutative formal group and applies it to cobordism theory.
We describe completely the link invariants constructed using Markov traces on the Yokonuma-Hecke algebras in terms of the linking matrix and the HOMFLYPT polynomials of sublinks.
Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to at fourth roots of unity, or by considering the super Hopf algebra . In this paper, we show …
Constructs Lagrangian skeleta for curve singularities.
problem Understanding Lagrangian skeleta of curve singularities.
method Constructs closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities.
result Provides computations of Legendrian and Weinstein invariants.
The study examines obstructions to links being shake slice.
problem Understanding when links are not shake slice.
method Examined shake concordance and zero surgery manifolds, and provided obstructions based on Arf invariants and algebraic sliceness.
result Links that are shake concordant have homology cobordant zero surgery manifolds, and provided specific obstructions to shake sliceness.
Defines braids with double lines for links in a surface times circle and connects it to the affine Hecke algebra.
problem Presenting links in a surface times circle using braids with double lines.
method Defines braids with double lines, proves Alexander and Markov theorems, and connects Hecke algebra to affine Hecke algebra.
result The Hecke algebra of braids with double lines is isomorphic to the affine Hecke algebra.
The study proves nearly Frobenius algebras over certain domains are Frobenius.
problem Understanding nearly Frobenius algebras and their properties.
method Analyzing nearly Frobenius algebras over principal ideal domains with specific algebraic properties.
result Any nearly Frobenius algebra with surjective multiplication and injective comultiplication is a Frobenius algebra.