Related papers: Fundamental group in the projective knot theory

The extension of the knot group $\pi_1(S^3\setminus K)$ to the category of tangles is introduced via a new category-theoretic construction. Through this presentation, a new avenue of proof for results about knot groups is opened.

We show that if two links in the real projective 3-space $\mathbb{RP}^{3}$ have isotopic preimages in the 3-sphere $S^{3}$ by the double covering map, then they are themselves isotopic in $\mathbb{RP}^{3}$.

A knot K is called n-adjacent to another knot K', if K admits a projection containing n generalized crossings such that changing any 0 < m \leq n of them yields a projection of K'. We apply techniques from the theory of sutured 3-manifolds,…

Let $p\neq 2$, and let $R$ be a smooth affine algebra of dimension $3$ over $\overline{F}_p$ and $P, Q$ be projective $R$-modules of rank $2$, each with trivial determinant. We prove: $P$ is isomorphic to $Q$ if and only if there is an…

For a link $L$ in the $3$-sphere, the $\pi$-orbifold group $G^\mathrm{orb}(L)$ is defined as a quotient of the link group $G(L)$ of $L$. When there exists an epimorphism $G^\mathrm{orb}(L)\to G^\mathrm{orb}(L')$ fitting into a certain…

We show that there are infinitely many triples of non-isotopic hyperbolic links in the lens space $L(4,1)$ such that the three lifts of each triple in $S^{3}$ are isotopic. They are obtained as the lifts of links in $S^{3} / Q_{8}$ by…

In this note we prove a decomposition related to the affine fundamental group and the projective fundamental group of a line arrangement and a reducible curve with a line component. We give some applications to this result.

We introduce the notion of the projective linking number Link(M,Z) of a compact oriented real submanifold M of dimension 2p-1 in complex projective n-space P^n with an algebraic subvariety Z in P^n - M of codimension p. This notion is…

The problem of whether different projectivizations of the same affine knot $K\subset\mathbb{S}^3$ are equivalent in $\mathbb{R}\mathbb{P}^3$ can be found in [11] and has also been posed as an open question in [15]. In this note we provide a…

We show that if a smooth projective curve $C\subset\mathbb P^3$ (over an algebraically closed field of characteristic zero) is Legendrian with respect to a contact structure (it is well known that a contact structure on $\mathbb P^3$ is…

Let $G$ be a compact, simply connected simple Lie group. We give a construction of an equivariant gerbe with connection on $G$, with equivariant 3-curvature representing a generator of $H^3_G(G,\Z)$. Technical tools developed in this…

In this paper we study rational real algebraic knots in $\R P^3$. We show that two real algebraic knots of degree $\leq5$ are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any irreducible…

We work on the notions of rail arcs and rail isotopy in $\mathbb{R}^3$, and we introduce the notions of rail knotoid diagrams and their equivalence. Our main result is that two rail arcs in $\mathbb{R}^3$ are rail isotopic if and only if…

A link of an isolated singularity of a two-dimensional semialgebraic surface in $R^4$ is a knot (or a link) in $S^3$. Thus the ambient Lipschitz classification of surface singularities in $R^4$ can be interpreted as a bi-Lipschitz…

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal…

The conormal Lagrangian $L_K$ of a knot $K$ in $\mathbb{R}^3$ is the submanifold of the cotangent bundle $T^* \mathbb{R}^3$ consisting of covectors along $K$ that annihilate tangent vectors to $K$. By intersecting with the unit cotangent…

Let $X$ be an irreducible smooth complex projective variety. Let $G$ be a linear algebraic group over $\mathbb{C}$. We define the notion of Lie algebroid valued connection on holomorphic principal $G$--bundles on $X$, and study their basic…

Let $X \subset \P^r$ be a linearly normal variety defined by a very ample line bundle $L$ on a projective variety $X$. Recently it is shown in \cite{HLMP} that there are many cases where $(X,L)$ satisfies property $\textsf{QR} (3)$ in the…

A pointwise-elliptic subset of a topological group is one whose elements all generate relatively-compact subgroups. A connected locally compact group has a dense pointwise-elliptic subgroup if and only if it is an extension by a compact…

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 1). We also consider an equivalence relation that is called…