Related papers: Pseudo links and singular links in the Solid Torus
It was proven by Gonz\'alez-Meneses, Manch\'on and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph…
The altenating knots, links and twists projected on the S_2 sphere are identified with the phase Space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossing points, the edges, to the…
In this paper we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side we state the relations between classical and virtual singular objects, in addition we discuss a Birman-like…
Physical knots and links are one-dimensional submanifolds of R^3 with fixed length and thickness. We show that isotopy classes in this category can differ from those of classical knot and link theory. In particular we exhibit a Gordian…
We construct infinite families of non-simple isotopy classes of links in overtwisted contact structures on $S^1$-bundles over surfaces. These examples include: (1) a pair of Legendrian links that are not Legendrian isotopic, but which are…
We introduce and study strongly invertible Legendrian links in the standard contact three-dimensional space. We establish the equivariant analogs of basic results separately well-known for strongly invertible and Legendrian links, i.e. the…
Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander-Markov correspondence for the theory of stable isotopy classes of…
In this paper we indicate one method of construction of linear representations of groups and algebras with translation invariant (except, maybe , finite number) defining relationships. As an illustration of this method, we give one approach…
In this paper we present a detailed study of \emph{bonded knots} and their related structures, integrating recent developments into a single framework. Bonded knots are classical knots endowed with embedded bonding arcs modeling physical or…
We present a complete classification of spherical knotoids with up to six crossings and conjecture that our classification up to seven crossings is complete. Our work extends the tradition of knot tabulation to the setting of knotoids…
Deformations of knots and links in ambient space can be studied combinatorially on their diagrams via local modifications called Reidemeister moves. While it is well-known that, in order to move between equivalent diagrams with Reidemeister…
We partially solve the conjecture by A.Shumakovitch about torsion in the Khovanov homology of prime, non-split links in S^3. We give a size restriction on the Khovanov homology of almost alternating links. We relate the Khovanov homology of…
We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki's…
Let $g_t$ be a loop in the space of monic complex polynomials in one variable of fixed degree $n$. If the roots of $g_t$ are distinct for all $t$, they form a braid $B_1$ on $n$ strands. Likewise, if the critical points of $g_t$ are…
A link in $S^3$ is called real algebraic if it is the link of an isolated singularity of a polynomial map from $\mathbb{R}^4$ to $\mathbb{R}^2$. It is known that every real algebraic link is fibered and it is conjectured that the converse…
The present paper is an addendum to "Spherical structures on torus knots and links", arXiv:1008.0312, and concerns more general case of torus knot and link cone-manifolds.
Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from potentially singular complex algebraic surfaces and complex curves inside them. We prove that knot lattice…
We introduce stable equivalence classes of oriented links in orientable three-manifolds that are orientation $I$-bundles over closed but not necessarily orientable surfaces. We call these twisted links, and show that they subsume the…
A method to obtain explicit and complete topological solution of SU(2) Chern-Simons theory on $S^3$ is developed. To this effect the necessary aspects of the theory of coloured-oriented braids and duality properties of conformal blocks for…
The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies to classical knot and classical…