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A parametric curve $\gamma$ of class $C^n$ on the $n$-sphere is said to be nondegenerate (or locally convex) when $\det\left(\gamma(t),\gamma'(t),\cdots,\gamma^{(n)}(t)\right)>0$ for all values of the parameter $t$. We orthogonalize this…
We report on some recent work on deformation of spaces, notably deformation of spheres, describing two classes of examples. The first class of examples consists of noncommutative manifolds associated with the so called $\theta$-deformations…
We consider $\mathbb{R}^3$ as a homogeneous manifold for the action of the motion group given by rotations and translations. For an arbitrary $\tau\in \widehat{SO(3)}$, let $E_\tau$ be the homogeneous vector bundle over $\mathbb{R}^3$…
We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On the level of Grothendieck groups this…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
Let $k$ be an algebraically closed field of characteristic $p > 3$ and $S$ be a smooth projective surface over $k$ with $k$-rational point $x$. For $n \geq 2$, let $S^{[n]}$ denote the Hilbert scheme of $n$ points on $S$. In this note, we…
We find further evidence for the conjecture relating large N Chern-Simons theory on S^3 with topological string on the resolved conifold geometry by showing that the Wilson loop observable of a simple knot on S^3 (for any representation)…
We define an invariant of triple-point-free immersions of $2$-spheres into Euclidean $3$-space, taking values in $l^1(\mathbb{Z})$. It remains unchanged under regular homotopies through such immersions. An explicit description of its image…
We first present three graphic surgery formulae for the degree $n$ part $Z_n$ of the Kontsevich-Kuperberg-Thurston universal finite type invariant of rational homology spheres. Each of these three formulae determines an alternate sum of the…
The fundamental quandle is a complete invariant for unoriented tame knots \cite{JO, Ma} and non-split links \cite{FR}. The proof involves proving a relationship between the components of the fundamental quandle and the cosets of the…
We prove some classification results for tight contact structure in the 3-space, -ball and -sphere that are invariant with respect to some arbitrary involution, that is conjugated to the standard rotation around the x-axis. Unlike the…
A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous…
The algebra of invariants of d-tuples of n x n skew-symmetric matrices under the action of the orthogonal group by simultaneous conjugation is considered over an infinite field of characteristic different from two. For n=3 and d>0 a minimal…
We investigate the rings of semi-invariants for tame string algebras A(n) of non-polynomial growth. We are interested in dimension vectors of band modules. We use geometric technique related to the description of coordinate rings on…
We study Seifert linking form which is an invariant of embeddings of punctured $n$-manifolds in $\mathbb R^{2n-1}$. For punctured $n$-manifold $N_0$ the values of this invariant are integer valued bilinear symmetric forms on…
This is the first paper in a general program to automate skein theoretic arguments. In this paper, we study skein theoretic invariants of planar trivalent graphs. Equivalently, we classify trivalent categories, which are nondegenerate…
We construct a simple topological invariant of certain 3-manifolds, including quotients of the 3-sphere by finite groups, based on the fact that the tangent bundle of an orientable 3-manifold is trivialisable. This invariant is strong…
We compute the factorisation homology of the four-punctured sphere and punctured torus over the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ explicitly as categories of equivariant modules using the framework of `Integrating Quantum…
We construct a state-sum type invariant of smooth closed oriented $4$-manifolds out of a $G$-crossed braided spherical fusion category ($G$-BSFC) for $G$ a finite group. The construction can be extended to obtain a $(3+1)$-dimensional…
We show that momentum-space tensor monopoles corresponding to nontrivial vector bundle generalizations, known as bundle gerbes, can be realized in bands of three-dimensional topological matter with nontrivial Hopf invariants. We provide a…