Related papers: Rail knotoids
In the present paper we construct a one-to-one correspondence between the set of graph-knots and the set of homotopy classes of looped graphs. Moreover, the graph-knot and the homotopy class constructed from a given knot are related with…
We realize a homological block of a knot complement in $S^3$ for $G_{\mathbb{C}}=SL(2,\mathbb{C})$ as a half-index of a 3d $\mathcal{N}=2$ theory via an expression of the homological block as an inverted Habiro series by working out some…
In this study of the Reidemeister moves within the classical knot theory, we focus on hard diagrams of knots and links, categorizing them as either rigid or shaky based on their adaptability to certain moves. We establish that every link…
Given a simply connected space $X$, there are several, a priori different, algebraic groups whose groups of $\mathbb Q$-points are isomorphic to the group of homotopy classes of homotopy automorphisms of the rationalization of $X$. We will…
Recently there had been a great deal of activity associated with various schemes of designing both analytical and experimental methods describing knotted structures in electrodynamics and in hydrodynamics.The majority of works in…
Grid diagrams encode useful geometric information about knots in S^3. In particular, they can be used to combinatorially define the knot Floer homology of a knot K in S^3, and they have a straightforward connection to Legendrian…
In this paper we investigate the relationship between isotopy classes of knots and links in S^3 and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the…
In 1974, D. Rolfsen asked: If two PL links in $S^3$ are isotopic (=homotopic through embeddings), then are they PL isotopic? We prove that they are PL isotopic to another pair of links which are indistinguishable from each other by finite…
The aims of this letter are three-fold: First is to show that nonlinear generalizations of electrodynamics support various types of knotted solutions in vacuum. The solutions are universal in the sense that they do not depend on the…
Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which,…
Classical knots in $\mathbb{R}^3$ can be represented by diagrams in the plane. These diagrams are formed by curves with a finite number of transverse crossings, where each crossing is decorated to indicate which strand of the knot passes…
The existence of ring-like and knotted solitons in O(3) non-linear sigma model is analysed. The role of isotopy of knots/links in classifying such solitons is pointed out. Appearance of torus knot solitons is seen.
We first prove the Grinberg-Kazhdan formal arc theorem without any assumptions on the characteristic. This part of the article is equivalent to arXiv:math-AG/0203263. Then we try to clarify the geometric ideas behind the proof by…
We prove that any link in $S^3$ whose Khovanov homology is the same as that of a Hopf link must be isotopic to that Hopf link. This holds for both reduced and unreduced Khovanov homology, and with coefficients in either $\mathbb{Z}$ or…
We discuss the relationship between two analogues in a 3-manifold of the set of prime ideals in a number field. We prove that if $(K_i)_{i\in \mathbb{N}_{>0}}$ is a sequence of knots obeying the Chebotarev law in the sense of Mazur and…
Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of…
We show that for every m in N, there exists an n in N such that every embedding of the complete graph K_n in R^3 contains a link of two components whose linking number is at least m. Furthermore, there exists an r in N such that every…
Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most ${2n\choose n}$ distinct minimal genus Heegaard splittings of $S^3\setminus\eta(K)$. These splittings can be divided…
We extend the notion of intersection graphs for knots in the theory of finite type invariants to string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graph, and show…
This paper focuses on the graphs in the Petersen family, the set of minor minimal intrinsically linked graphs. We prove there is a relationship between algebraic linking of an embedding and knotting in an embedding. We also present a more…