Related papers: On Farber's invariants for simple $2q$-knots

This article presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we…

We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group $\pi$ such that $c.d.\pi=2$ and $\pi$ has one end. Let $\beta=\beta_2(\pi;F_2)$ and $w=w_1(X)$. Our main result is that (modulo two technical conditions on…

This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \pi_K be…

Using the skew-Hopf pairing, we obtain $\mathcal{R}$-matrix for the two-parameter quantum algebra $U_{v,t}$. We further construct a strict monoidal functor $\mathcal{T}$ from the tangle category $(\mathrm{OTa},\otimes, \emptyset)$ to the…

We recapture Kuperberg's numerical invariant of 3-manifolds associated to a semisimple and cosemisimple Hopf algebra through a `planar algebra construction'. A result of possibly independent interest, used during the proof, which relates…

Exceptional points at which eigenvalues and eigenvectors of non-Hermitian matrices coalesce are ubiquitous in the description of a wide range of platforms from photonic or mechanical metamaterials to open quantum systems. Here, we introduce…

Motivated by the Lagrange top coupled to an oscillator, we consider the quasi-periodic Hamiltonian Hopf bifurcation. To this end, we develop the normal linear stability theory of an invariant torus with a generic (i.e., non-semisimple)…

Using the Harpaz-Nuiten-Prasma interpretation of the Dwyer-Kan-Smith cohomology of a simplicial category $\mathcal{X}$, we obtain a cochain complex for the Andr\'{e}-Quillen cohomology groups in which the $k$-invariants for $\mathcal{X}$…

We build model structures on the category of equivariant simplicial operads with weak equivalences determined by families of subgroups, in the context of operads with a varying set of colors (and building on the fixed color model structures…

Quantum knot invariants (like colored HOMFLY-PT or Kauffman polynomials) are a distinguished class of non-perturbative topological invariants. Any known way to construct them (via Chern-Simons theory or quantum R-matrix) starts with a…

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…

In the present paper we study some homotopy invariants which can be defined by means of bundles with fiber a matrix algebra. We also introduce some generalization of the Brauer group in the topological context and show that any its element…

Kauffman's bracket is an invariant of regular isotopy of knots and links which since its discovery in 1985 it has been used in many different directions: (a) it implies an easy proof of the invariance of (in fact, it is equivalent to) the…

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel…

We construct a new type of quantum invariant of closed framed $3$-manifolds with the vanishing first Betti number. The invariant is defined for any finite dimensional Hopf algebra, such as small quantum groups, and is based on ideal…

We define and study an equivariant version of Farber's topological complexity for spaces with a given compact group action. This is a special case of the equivariant sectional category of an equivariant map, also defined in this paper. The…

By considering a particular type of invariant Seifert surfaces we define a homomorphism $\Phi$ from the (topological) equivariant concordance group of directed strongly invertible knots $\widetilde{\mathcal{C}}$ to a new equivariant…

The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum…

For a smooth manifold A, we consider the ordered configuration space F_k(AxR) of k distinct points in AxR. We obtain an explicit homotopy construction of the configuration space F_k(AxR) and of the (k-2)-fold suspension of F_k(A). Under…

We present an algorithm that, given finite simplicial sets $X$, $A$, $Y$ with an action of a finite group $G$, computes the set $[X,Y]^A_G$ of homotopy classes of equivariant maps $\ell \colon X \to Y$ extending a given equivariant map $f…