Related papers: Invariants of long knots
Link/knot invariants are series with integer coefficients, and it is a long-standing problem to get them positive and possessing cohomological interpretation. Constructing positive "superpolynomials" is not straightforward, especially for…
We study Hopf algebras via tools from geometric invariant theory. We show that all the invariants we get can be constructed using the integrals of the Hopf algebra and its dual together with the multiplication and the comultiplication, and…
There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional…
We construct an algebra of non-trivial homological operations on Khovanov homology with coefficients in $\mathbb Z_2$ generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift…
This paper extends the construction of invariants for virtual knots to virtual long knots and introduces two new invariant modules of virtual long knots. Several interesting features are described that distinguish virtual long knots from…
We introduce an invariant of a hyperbolic knot which is a map $\alpha\mapsto \boldsymbol{\Phi}_\alpha(h)$ from $\mathbb{Q}/\mathbb{Z}$ to matrices with entries in $\overline{\mathbb{Q}}[[h]]$ and with rows and columns indexed by the…
We define invariants for a framed link equipped with a SL2 local system in its complement and additional combinatorial data based on the theory of representations of stated skein algebras at roots of unity of punctured bigons and the…
We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors ?^{**} and D tensor ^{**}? tensor D^{-1}. This…
For a given group $G$, we construct an invariant of flat $G$-connections on 4-manifolds from a finite type involutory quasitriangular Hopf $G$-algebra. Hopf $G$-algebras are generalizations of Hopf algebras, equipped with gradings by $G$.…
We present a definition of an invariant #(M,H), defined for every finite-dimensional Hopf algebra (or Hopf superalgebra or Hopf object) H and for every closed, framed 3-manifold M. When H is a quantized universal enveloping algebra, #(M,H)…
In this paper, we show an isomorphism of homological knot invariants categorifying the Reshetikhin-Turaev invariants for $\mathfrak{sl}_n$. Over the past decade, such invariants have been constructed in a variety of different ways, using…
We prove exact complexity dichotomies for two quantum invariants of closed oriented three-manifolds, with the categorical data fixed. For a modular category $\mathcal{C}$, computing the Reshetikhin--Turaev invariant $Z_{\mathcal{C}}(M)$…
The colored Jones polynomial associated to a knot admits an expansion of knot invariants known as the large-color expansion or Melvin-Morton-Rozansky expansion. We will show how this expansion can be derived from the universal invariant…
In a previous work by the author it was shown that every finite dimensional algebraic structure over an algebraically closed field of characteristic zero K gives rise to a character $K[X]_{aug}\to K$, where $K[X]_aug$ is a commutative Hopf…
We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are…
We introduce topological invariants of knots and braid conjugacy classes, in the form of differential graded algebras, and present an explicit combinatorial formulation for these invariants. The algebras conjecturally give the relative…
The WRT invariant of a link L in S2xS1 at sufficiently high values of the level r can be expresses as an evaluation of a special polynomial invariant of L at 2r-th root of unity. We categorify this polynomial invariant by associating to L a…
We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the…
If H is a Hopf algebra with bijective antipode and \alpha, \beta \in Aut_{Hopf}(H), we introduce a category_H{\cal YD}^H(\alpha, \beta), generalizing both Yetter-Drinfeld and anti-Yetter-Drinfeld modules. We construct a braided T-category…
Given a group endowed with a Z/2-valued morphism we associate a Gauss diagram theory, and show that for a particular choice of the group these diagrams encode faithfully virtual knots on a given arbitrary surface. This theory contains all…