An invariant of tangle cobordisms

Double Braidings, Twists, and Tangle Invariants

A tortile (or ribbon) category defines invariants of ribbon (framed) links and tangles. We observe that these invariants, when restricted to links, string links, and more general tangles which we call turbans, do not actually depend on the braiding of the tortile category. Besides duality, the only pertinent data for such tangles are the double braiding and twist. We introduce the general notions of twine, which is meant to play the role of the double braiding (in the absence of a braiding), and the corresponding notion of twist. We show that the category of (ribbon) pure braids is the free ca…

A cobordism category attached to Khovanov-Rozansky link homologies based on operads

We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams with values in the homotopy category attached to the cobordism category. Motivated by Bar-Natan's categorification of the Jones polynomial, it categorifies the quantum $sl_n$ quantum invariants and is adapted to the categorification of the $sl_n$ quantum invariants by Khovanov and Rozansky using matrix factorizations. We conjecture to exist th…

Oriented Quantum Algebras and Coalgebras, Invariants of Oriented 1-1 Tangles, Knots and Links

In this paper we study oriented quantum coalgebras which are structures closely related to oriented quantum algebras. We study the relationship between oriented quantum coalgebras and oriented quantum algebras and the relationship between oriented quantum coalgebras and quantum coalgebras. We show that there are regular isotopy invariants of oriented 1-1 tangles and of oriented knots and links associated to oriented and twist oriented quantum coalgebras respectively. There are many parallels between the theory of oriented quantum coalgebras and the theory of quantum coalgebras

A functorial LMO invariant for Lagrangian cobordisms

Lagrangian cobordisms are three-dimensional compact oriented cobordisms between once-punctured surfaces, subject to some homological conditions. We extend the Le-Murakami-Ohtsuki invariant of homology three-spheres to a functor from the category of Lagrangian cobordisms to a certain category of Jacobi diagrams. We prove some properties of this functorial LMO invariant, including its universality among rational finite-type invariants of Lagrangian cobordisms. Finally, we apply the LMO functor to the study of homology cylinders from the point of view of their finite-type invariants.

Alexander type invariants of tangles

We study generalizations of a classical link invariant -- the multivariable Alexander polynomial -- to tangles. The starting point is Archibald's tMVA invariant for virtual tangles which lives in the setting of circuit algebras, and whose target space has dimension that is exponential in the number of strands. Using the Hodge star map and restricting to tangles without closed components, we define a reduction of the tMVA to an invariant "rMVA" which is valued in matrices with Laurent polynomial entries, and so has a much more compact target space. We show the rMVA has the structure of a metamo…

On the relation between two quantum group invariants of 3-cobordisms

We prove in the context of quantum groups at even roots of unity that a Turaev-Viro type invariant of a 3-dimensional cobordism M equals the tensor product of the Reshetikhin-Turaev invariants of M and M*, where M* denotes M with orientation reversed.

Alexander-Conway invariants of tangles

We consider an algebra of (classical or virtual) tangles over an ordered circuit operad and introduce Conway-type invariants of tangles which respect this algebraic structure. The resulting invariants contain both the coefficients of the Conway polynomial and the Milnor's mu-invariants of string links as partial cases. The extension of the Conway polynomial to virtual tangles satisfies the usual Conway skein relation and its coefficients are GPV finite type invariants. As a by-product, we also obtain a simple representation of the braid group which gives the Conway polynomial as a certain twis…

Diassociative algebras and Milnor's invariants for tangles

We extend Milnor's mu-invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for mu-invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves corresponds to axioms of Loday's diassociative algebra. The relation of tangles to diassociative algebras is formulated in terms of a morphism of corresponding operads.

On the universal sl_2 invariant of ribbon bottom tangles

A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are on a line in the bottom square of the cube. A ribbon bottom tangle is a bottom tangle whose closure is a ribbon link. For every n-component ribbon bottom tangle T, we prove that the universal invariant J_T of T associated to the quantized enveloping algebra U_h(sl_2) of the Lie algebra sl_2 is contained in a certain Z[q,q^{-1}]-subalgebra of the n-fold completed tensor power of U_h(sl_2). This result is applied to the colored Jones polynomial of ribbon links.

On the universal sl_2 invariant of boundary bottom tangles

The universal sl_2 invariant of bottom tangles has a universality property for the colored Jones polynomial of links. Habiro conjectured that the universal sl_2 invariant of boundary bottom tangles takes values in certain subalgebras of the completed tensor powers of the quantized enveloping algebra U_h(sl_2) of the Lie algebra sl_2. In the present paper, we prove an improved version of Habiro's conjecture. As an application, we prove a divisibility property of the colored Jones polynomial of boundary links.

The diagram category of framed tangles and invariants of quantized symplectic group

In this paper we present a categorical version of the first and second fundamental theorems of the invariant theory for the quantized symplectic groups. Our methods depend on the theory of braided strict monoidal categories which are pivotal, more explicitly the diagram category of framed tangles.

On the universal sl_2 invariant of Brunnian bottom tangles

A link L is called Brunnian if every proper sublink of L is trivial. Similarly, a bottom tangle T is called Brunnian if every proper subtangle of T is trivial. In this paper, we give a small subalgebra of the n-fold completed tensor power of U_h(sl_2) in which the universal sl_2 invariant of n-component Brunnian bottom tangles takes values. As an application, we give a divisibility property of the colored Jones polynomial of Brunnian links.

Ribbon Yetter--Drinfeld modules and tangle invariants

We define notions of pivotal and ribbon objects in a monoidal category. These constructions give pivotal or ribbon monoidal categories from a monoidal category which is not necessarily with duals. We apply this construction to the braided monoidal category of Yetter--Drinfeld modules over a Hopf algebra. This gives rise to the notion of ribbon Yetter--Drinfeld modules over a Hopf algebra, which form ribbon categories. This gives an invariant of tangles.

XC-结和通用不变量

XC-tangles and universal invariants

XC-结提供了一种研究量子不变量的框架，并扩展了Reshetikhin-Turaev函子，同时探讨了有限类型不变量理论。

Categorified sl(N) invariants of colored rational tangles

We use categorical skew Howe duality to find recursion rules that compute categorified sl(N) invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress towards two conjectures about the colored HOMFLY homology of rational links.

Bottom tangles and universal invariants

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 B, which provides an especially convenient way to generate all the bottom tangles, and hence all the framed, oriented links, via closure. We also define a kind of "braided Hopf algebra action" on the set of bottom tangles. Using the universal invariant of bottom tangles associat…

Khovanov invariants via Fukaya categories: the tangle invariants agree

Given a pointed 4-ended tangle $T \subset D^3$, there are two Khovanov theoretic tangle invariants, $\unicode{1044}_1(T)$ from [arXiv:1910.1458] and $L_T$ from [arXiv:1808.06957], which are twisted complexes over the Fukaya category of the boundary 4-punctured sphere $(S^2,4\text{pt})=\partial (D^3, T)$. We prove that these two invariants are the same.

An invariant of link cobordisms from Khovanov homology

In [Duke Math. J. 101 (1999) 359-426], Mikhail Khovanov constructed a homology theory for oriented links, whose graded Euler characteristic is the Jones polynomial. He also explained how every link cobordism between two links induces a homomorphism between their homology groups, and he conjectured the invariance (up to sign) of this homomorphism under ambient isotopy of the link cobordism. In this paper we prove this conjecture, after having made a necessary improvement on its statement. We also introduce polynomial Lefschetz numbers of cobordisms from a link to itself such that the Lefschetz …

一个精细化的函子性万有环带不变量

A refined functorial universal tangle invariant

本文构造了一个精细化的函子性万有环带不变量，通过XC-代数实现，保持辫子、扭转和开迹，细化了现有函子性不变量。

Quantum Enhancements of Involutory Birack Counting Invariants

The involutory birack counting invariant is an integer-valued invariant of unoriented tangles defined by counting homomorphisms from the fundamental involutory birack of the tangle to a finite involutory birack over a set of framings modulo the birack rank of the labeling birack. In this first of an anticipated series of several papers, we enhance the involutory birack counting invariant with quantum weights, which may be understood as tangle functors of involutory birack-labeled unoriented tangles.

Cyclotomic associators and finite type invariants for tangles in the solid torus

The universal Vassiliev-Kontsevich invariant is a functor from the category of tangles to a certain graded category of chord diagrams, compatible with the Vassiliev filtration and whose associated graded is an isomorphism. The Vassiliev filtration has a natural extension to tangles in any thickened surface $M\times I$ but the corresponding category of diagrams lacks some finiteness properties which are essential to the above construction. We suggest to overcome this obstruction by studying families of Vassiliev invariants which, roughly, are associated to finite coverings of $M$. In the case $…

A Diagrammatic Multivariate Alexander Invariant of Tangles

Recently, Bigelow defined a diagrammatic method for calculating the Alexander polynomial of a knot or link by resolving crossings in a planar algebra. I will present my multivariate version of Bigelow's calculation. The advantage to my algorithm is that it generalizes to a multivariate tangle invariant up to Reidemeister I. I will conclude with a possible link to subfactor planar algebras from the work of Jones and Penneys.

A Categorification of Quantum sl_3 Projectors and the sl_3 Reshetikhin-Turaev Invariant of Tangles

We construct a categorification of the quantum sl_3 projectors, the sl_3 analog of the Jones-Wenzl projectors, as the stable limit of the complexes assigned to k-twist torus braids (as k goes to infinity) in a suitably shifted version of Morrison and Nieh's geometric formulation of sl_3 link homology (math.GT/0612754). We use these projectors to give a categorification of the sl_3 Reshetikhin-Turaev invariant of framed tangles.

A cohomology theory for colored tangles

We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labelled by irreducible representations of U_q(sl(2)). We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the corresponding theory is a categorification of the colored Jones polynomial, and provides a tool for efficient computations of the resulting colored invariant of knots and links. Our theory is defined over the Gaussian integers Z[i] (and more generally over Z[i][a,h], where a,h ar…

Categorification: tangle invariants and TQFTs

Based on different views on the Jones polynomial we review representation theoretic categorified link and tangle invariants. We unify them in a common combinatorial framework and connect them via the theory of Soergel bimodules. The influence of these categorifications on the development of 2-representation theory and the interaction between topological invariants and 2-categorical structures is discussed. Finally, we indicate how categorified representations of quantum groups on the one hand and monoidal 2-categories of Soergel bimodules on the other hand might lead to new interesting 4-dimen…

Invariants of tangles with flat connections in their complements

Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide examples of G-categories.

Tangle blocks in the theory of link invariants

The central discovery of $2d$ conformal theory was holomorphic factorization, which expressed correlation functions through bilinear combinations of conformal blocks, which are easily cut and joined without a need to sum over the entire huge Hilbert space of states. Somewhat similar, when a link diagram is glued from tangles, the link polynomial is a multilinear combination of {\it tangle blocks} summed over just a few representations of intermediate states. This turns to be a powerful approach because the same tangles appear as constituents of very different knots so that they can be extracte…

SL₂(C) 陈-维特齐斯不变量的量子化

A quantization of the $\operatorname{SL}_2(\mathbb{C})$ Chern-Simons invariant of tangle exteriors

本文定义了tangles的平坦sl₂连接不变量Z_N^ψ，作为Kashaev不变量的几何扭曲或SL₂(C)陈-维特齐斯不变量的量子化。通过新的描述I^ψ验证了后者，并展示了Z_N^ψ在N=1时直接恢复I^ψ。

Tangle addition and the knots-quivers correspondence

We prove that the generating functions for the one row/column colored HOMFLY-PT invariants of arborescent links are specializations of the generating functions of the motivic Donaldson-Thomas invariants of appropriate quivers that we naturally associate with these links. Our approach extends the previously established tangles-quivers correspondence for rational tangles to algebraic tangles by developing gluing formulas for HOMFLY-PT skein generating functions under Conway's tangle addition. As a consequence, we prove the conjectural links-quivers correspondence of Kucharski-Reineke-Stošić-Sulk…

缠结的 Chen-Khovanov 谱

Chen-Khovanov spectra for tangles

本文指出 Khovanov 弧代数与缠结不变量的稳定同伦细化可诱导 Chen-Khovanov 和 Stroppel 平台代数及缠结不变量的细化，并研究其拓扑 Hochschild 同调。

Skein theories for virtual tangles

In this paper, we use skein-theoretic techniques to classify all virtual knot polynomials and trivalent graph invariants with certain smallness conditions. The first half of the paper classifies all virtual knot polynomials giving non-trivial invariants strictly smaller than the one given by the Higman-Sims spin model. In particular, we exhibit a family of skein theories coming from $\text{Rep}(O(2))$ with an interesting braiding. In addition, all skein theories of oriented virtual tangles with some smallness conditions are classified. In the second half of the paper, we classify all non-trivi…

Annular Khovanov-Lee homology, braids, and cobordisms

We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type is an invariant of the isotopy class of L in A x I. Using ideas of Ozsvath-Stipsicz-Szabo as reinterpreted by Livingston, we use this structure to define a family of annular Rasmussen invariants that yield information about annular and non-annular cobordisms. Focusing on the special case of annular links obtained as braid closures, we use the behavior of the annular Rasmussen invariants to obtain a necessary condition for bra…

双缠结的Heegaard Floer多重曲线

Heegaard Floer multicurves of double tangles

本文给出了从纽结补集Heegaard Floer多重曲线计算双缠结不变量的公式，并借此发现纽结Floer同调与Khovanov同调在卫星运算下行为迥异，同时给出L空间纽结的新刻画及首个Floer同调为薄的卫星纽结。

On the Connectivity of Cobordisms and Half-Projective TQFT's

We consider a generalization of the axioms of a TQFT, so called half-projective TQFT's, with an anomaly, $x^μ$, in the composition law. $μ$ is a coboundary on the cobordism categories with non-negative, integer values. The element $x$ of the ring over which the TQFT is defined does not have to be invertible. In particular, it may be 0. This modification makes it possible to extend quantum-invariants, which vanish on $S^1\times S^2$, to non-trivial TQFT's. (A TQFT in the sense of Atiyah with this property has to be trivial all together). Under a few natural assumptions the notion of a half-proj…