The paper introduces a cobordism for Khovanov homology crossing change and categorifies Vassiliev skein relations.
problem Khovanov homology and crossing changes in tangle diagrams.
method Introducing a sum of cobordisms that yields a morphism on Khovanov homology complexes for crossing change.
result The introduced cobordism is invariant under double point moves and categorifies Vassiliev skein relations.
New method computes first Vassiliev derivative of Khovanov homology.
problem Computing Vassiliev derivatives of Khovanov homology.
method Developed a crux complex to compute the first derivative.
result Direct computation of the first derivative of Khovanov homology.
Homflypt skein theory and string topology linked via 2-groupoids.
problem Understanding relations in Homflypt skein theory.
method Defined a 2-groupoid from the fundamental 2-groupoid of singular links, relating it to string topology.
result Relations in Homflypt skein theory are induced from a 2-groupoid.
Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this …
The study extends knot theory to knotoids using two approaches.
problem Extending Vassiliev invariants to knotoids.
method Two approaches: 1) Closures to knots, 2) Directly on knotoids.
result Non-trivial type-1 invariants for spherical knotoids.
Explains basic knot and link theory for students.
problem Understanding simple invariants of knots and links.
method Defining and explaining linking number, Arf-Casson invariants, Alexander-Conway polynomials, and skein relations.
result Introduction of stronger invariants and the Vassiliev-Kontsevich theorem.
Pulling back the weight system associated with the exceptional Lie algebra G_2 by a modification of the universal Vassiliev-Kontsevich invariant yields a link invariant; extending it to 3-nets, we derive a recursive algorithm for its evaluation.
The paper calculates a specific weight system for chord diagrams with a particular graph structure.
problem Calculating a specific weight system for chord diagrams with a complete bipartite graph structure.
method Using a Lie algebra sl3 and its weight system, the authors derive a function on chord diagrams. result The authors compute the sl3 weight system for chord diagrams with a complete bipartite graph structure. In this paper we define and present a simple combinatorial formula for a 3-variable Laurent polynomial invariant of conjugacy classes in Artin braid group Bm. We show that this Laurent polynomial satisfies the Conway skein relation and its coefficients are Vassiliev invariants of braids.
New skein theory for Links-Gould polynomial simplifies link evaluations.
problem Computing Links-Gould polynomial for oriented links.
method Developed a cubic braid-type skein theory.
result Skein theory can evaluate any oriented link.
Pulling back the weight system associated with the spinor representation of the Lie algebra so(7) by the universal Vassiliev-Kontsevich invariant yields a numerical link invariant with values in formal power series. Computing some skein relations satisfied by this invariant, I derive a recursive algorithm for its evalu…
In~\cite{Kim} the author generalized the Conway algebra and constructed the invariant valued in the generalized Conway algebra defined by applying two skein relations to crossings, which is called a generalized Conway type invariant. The generalized Conway type invariant is a generalization of Homflypt polynomial. In t…
We study generalizations of finite-type knot invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move, analyzing in detail the band-pass and doubled-delta moves. Using braid-theoretic techniques, we show that, for a large class of local moves, generalized Goussarov's…
Study algebraic relations of Vassiliev invariants for families of knots.
problem Understanding algebraic structure of Vassiliev invariants for knot families.
method Analyzing algebraic relations and generating sets of Vassiliev invariants in 3D Chern-Simons theory.
result For 1-parametric knot families, Vassiliev invariants are finitely generated. For more parameters, there can be an infinite number of generators.
Proves a categorified relation in Khovanov homology.
problem None explicitly stated; focuses on categorification of a relation.
method Categorified analogue of Kontsevich's 4T relation on Khovanov homology.
result Proof of categorified 4T relation in Khovanov homology.
Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case - the celebrated Casson-Walker invariant of rational homology spheres. This pap…
A formula for the difference of Vassiliev invariants of degree k+1 of two knots all of whose Vassiliev invariants of degree k agree is proven. The proof uses K. Habiro's C-moves and his theorem which relates them to Vassiliev invariants.
The paper explores basic properties of knot skein invariants.
problem Understanding skein invariants of knots.
method Discussion of basic properties and known examples of skein invariants.
result Discussion of basic properties and known examples of skein invariants.
Proves a pentagon relation in skein theory.
problem Developed a skein-theoretic version of cluster theory and conjectured a pentagon relation.
method Topological proof using skein algebra and elliptic Hall algebra.
result Proves the pentagon relation for the skein dilogarithm.
Paper compares skein modules to Kauffman bracket modules.
problem Comparing skein modules to Kauffman bracket modules.
method Using skein relations and Reshetikhin-Turaev model.
result Resolved the problem of comparing skein modules to Kauffman bracket modules.
We introduce the deformed fermionic numbers, corresponding to the skein relations, the main characteristics of knots and links. These fermionic numbers allow one to restore the skein relations. For the Alexander (Jones) skein relation we introduce corresponding Alexander (Jones) fermionic q-numbers, and for the HOMFLY …
Proves partial-dual genus polynomial is a knot invariant weight system.
problem Proving the partial-dual genus polynomial is a weight system.
method Proved the polynomial satisfies the four-term relation, thus making it a weight system.
result The partial-dual genus polynomial is a Vassiliev knot invariant weight system.
This paper is an exposition of heuristics related to Witten's functional integral, relating it to Vassiliev invariants and to the Kontsevich integrals that can be used to produce Vassiliev invariants of knots and links.In particular, we give a simplified version of the appearance of the Kontsevich integrals in the pert…
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.
Paper introduces a new skein relation for multivariable polynomials of virtual links.
problem Understanding properties of virtual links through polynomial invariants.
method Developed a virtual skein relation for multivariable polynomials of virtual links.
result New skein relation for multivariable polynomials of virtual links.
Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.
problem Understanding the skein-valued cluster transformation in Legendrian surfaces.
method Geometric considerations of moduli of holomorphic curves.
result Skein-valued cluster transformation of Legendrian surfaces related by surgery.
Relates two types of skein algebras using explicit correspondences.
problem Defining and relating stated and internal skein algebras.
method Explicit correspondence between stated and internal skein algebras, distinguishing between left and right boundary edges, proving excision properties.
result Agrees with excision properties of stated skein algebras under specific conditions.
Paper derives matrix formulae and proves skein relations for non-orientable surfaces in quasi-cluster algebras.
problem Understanding quasi-cluster algebras on non-orientable surfaces.
method Developed matrix formulae and proved skein relations for quasi-cluster variables.
result Laurent expansion and skein relations for quasi-cluster variables on non-orientable surfaces.
Defines Vassiliev complexity measures for open and closed curves in 3D space.
problem Measuring complexity of curves in 3D space.
method Using enhanced Jones polynomial coefficients and Gauss code diagrams.
result Second Vassiliev measure converges to knot invariants as curve ends coincide.
The theory of link-homotopy, introduced by Milnor, is an important part of the knot theory, with Milnor's mu-bar-invariants being the basic set of link-homotopy invariants. Skein relations for knot and link invariants played a crucial role in the recent developments of knot theory. However, while skein relations for Al…
Quantum theory constructs a group and skein module for knot complements.
problem Understanding the fundamental group of knot complements using quantum methods.
method Using bottom tangles, the universal space of quantum representations is constructed, then factored by the skein relation to get the skein module.
result Derives recurrence relation for the colored Jones polynomial, known as Aq polynomial. For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skei…
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…
Introduces admissible skein modules for non-semisimple categories.
problem No specific problem stated; generalization of Kauffman skein algebra.
method Introduces admissible skein modules associated to ideals in pivotal categories.
result These modules generalize Kauffman skein algebra and relate to quantum invariants.
The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's G2 quantum invariants. Our method consists in the study of Mark…
In this note we define a polynomial invariant for colored links by a skein relation. It specializes to the Jones polynomial for classical links.
Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulate…
The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are ab…
A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is …
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a given invariant. A skein invariant can be computed on each link solely by the use…
Study of skein invariants on tori for various groups and quantum parameters.
problem Analysis of G-skein theory invariants on tori for different groups and parameters. method Combinatorial and algebraic methods, including DAHA and skein relations.
result Isomorphisms and homomorphisms between skein algebras and DAHA, proving equivalence of tangles.
The paper calculates a new invariant for 4-manifolds using handle decompositions and skein relations.
problem Computing invariants for 4-manifolds built from handles.
method Handle attachment formulas, cabled colimits, lasso relation.
result Explicit calculations and partial vanishing results for specific 4-manifolds.
Researchers compute the Kauffman bracket skein module of a specific 3-manifold.
problem Understanding the structure of Kauffman bracket skein modules for non-prime manifolds.
method Analyzing handle sliding relations to compute the module over Z[A±1]. result The skein module of (S1imesS2) # (S1imesS2) does not split into free and torsion submodules. Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a system of skein equations to construct link invariant. This invariant can also be …
We define a 1-cocycle in the space of long knots that is a natural generalization of the Kontsevich integral seen as a 0-cocycle. It involves a 2-form that generalizes the Knizhnik--Zamolodchikov connection. We show that the well-known close relationship between the Kontsevich integral and Vassiliev invariants (via the…
In view of the result of Kontsevich, now often called ``the fundamental theorem of Vassiliev theory'', identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the ``4T-relation'' (and in the unframed case, th…
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.
As one of the problems in his list [20], T. Ohtsuki proposed to study relations between quandle cocycle invariants and quantum invariants. The aim of this paper is to answer one of those questions. We prove that the coefficient of the finite perturbative expansion of the quandle shadow cocycle invariant defined by $(\Z…