Related papers: Simple expressions for the holed torus relations

We describe polar homology groups for complex manifolds. The polar k-chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincare residue…

Let $N_g^k$ be a nonorientable surface of genus \ $g\geq 5$ \ with \ $k$-punctures. In this note, we will give an algebraic characterization of a Dehn twist about a simple closed curve on $N_g^k$. Along the way, we will fill some little…

Let G be a connected linear algebraic group over a field k. We say that G is toric-friendly if for any field extension K/k and any maximal K-torus T in G the group G(K) has only one orbit in (G/T)(K). Our main result is a classification of…

Using factorization homology techniques, we give a simple proof of the action of Kontsevich and Soibelman operad on the pair consisting of Hochschild cohomology and Hochschild homology. The proof obviously extends to higher Hochschild…

We show that for any nontrivial knot $K$ and any natural number $n$ there is a diagram $D$ of $K$ such that the unknotting number of $D$ is greater than or equal to $n$. It is well known that twice the unknotting number of $K$ is less than…

We give explicit realizations with small integer coordinates for all triangulated tori with up to 12 vertices. In particular, we provide coordinate-minimal realizations in general position for all triangulations of the torus with 7, 8, 9,…

Let $\overline G$ be the wonderful compactification of a simple affine algebraic group $G$ defined over $\mathbb C$ such that its center is trivial and $G\not= {\rm PSL}(2,\mathbb{C})$. Take a maximal torus $T \subset G$, and denote by…

Working over an algebraically closed field $k$ of any characteristic, we determine the matrix factorizations for the --- suitably graded --- triangle singularities $f=x^a+y^b+z^c$ of domestic type, that is, we assume that $(a,b,c)$ are…

We describe a class $\mathcal{C}$ of punctured torus bundles such that, for each $M \in \mathcal{C}$, all but finitely many Dehn fillings on $M$ are virtually Haken. We show that $\mathcal{C}$ contains infinitely many commensurability…

We classify all Hopf algebras which factorize through two Taft algebras $\mathbb{T}_{n^{2}}(\bar{q})$ and respectively $T_{m^{2}}(q)$. To start with, all possible matched pairs between the two Taft algebras are described: if $\bar{q} \neq…

A matrix factorization problem is considered. The matrix to be factorized is algebraic, has dimension 2 X 2 and belongs to Moiseev's class. A new method of factorization is proposed. First, the matrix factorization problem is reduced to a…

The Harer-Zagier (HZ) transform maps the HOMFLY-PT polynomial into a rational function. For some special knots and links, the latter admits a simple factorised form, which is referred to as HZ factorisation. This property is preserved under…

In this work the matrix exponential function is solved analytically for the special orthogonal groups $SO(n)$ up to $n=9$. The number of occurring $k$-th matrix powers gets limited to $0\leq k \leq n-1$ by exploiting the Cayley-Hamilton…

We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds…

We prove lower bounds for the fraction of edges of an $r$-graph which can be covered by the union of $k$ 1-factors. The special case $r=3$ yields some known results for cubic graphs. Furthermore, we introduce the concept of…

We study complete, simply-connected manifolds with special holonomy that are toric with respect to their multi-moment maps. We consider the cases where there is a connected non-Abelian symmetry group containing the torus. For…

A relation on a $k$-net$(n)$ (or, equivalently, a set of $k-2$ mutually orthogonal Latin squares of order $n$) is an $\mathbb{F}_{2}$ linear dependence within the incidence matrix of the net. Dukes and Howard (2014) showed that any…

Knotted vortices such as those produced in water by Kleckner and Irvine tend to transform by reconnection to collections of unknotted and unlinked circles. The reconnection number $R(K)$ of an oriented knot of link $K$ is the least number…

In 2000, Thomas Fink and Young Mao studied neck ties and, with certain assumptions, found 85 different ways to tie a neck tie. They gave a formal language which describes how a tie is made, giving a sequence of moves for each neck tie. The…

We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid…