Related papers: From annular to toroidal pseudo knots
We present those properties of planar doodles, especially when regarded as 4-valent graphs, that enable us to classify them into {\it prime} and {\it super prime} doodles by analogy to a knot sum. We describe a method for partially…
We study collections of planar curves that yield diagrams for all knots. In particular, we show that a very special class called potholder curves carries all knots. This has implications for realizing all knots and links as special types of…
We present a direct connection between torus knots and Hopfions by finding stable and static solutions of the extended Faddeev-Skyrme model with a ferromagnetic potential term. (P,Q)--torus knots consisting of |Q| sine-Gordon kink strings…
This is an introductory article on high dimensional knots for the beginners. High dimensional knot theory is an exciting field. It is a field of knot theory, which is one of topology and is connected with many ones. In this article we use…
We define an annular concordance invariant and study its properties. When specialized to braids, this invariant gives bounds on band rank. We introduce a modified chain complex to reformulate the invariant. Then, by focusing on a special…
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other…
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…
Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.
In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the…
Let $K$ be a genus $g$ alternating knot with Alexander polynomial $\Delta_K(T)=\sum_{i=-g}^ga_iT^i$. We show that if $|a_g|=|a_{g-1}|$, then $K$ is the torus knot $T_{2g+1,\pm2}$. This is a special case of the Fox Trapezoidal Conjecture.…
Extending a proposal of Defant and Kravitz [Discrete Mathematics, \textbf{1}, 347 (2024)], we define Hitomezashi patterns and loops on a torus and provide several structural results for such loops. For a given pattern, our main theorems…
Simple closed curves in the plane can be mapped to nontrivial knots under the action of origami foldings that allow the paper to self-intersect. We show all tame knot types may be produced in this manner, motivating the development of a new…
We introduce shadow structures for singular knot theory. Precisely, we define \emph{two} invariants of singular knots and links. First, we introduce a notion of action of a singquandle on a set to define a shadow counting invariant of…
This paper discusses a generalization of virtual knot theory that we call multi-virtual knot theory. Multi-virtual knot theory uses a multiplicity of types of virtual crossings. As we will explain, this multiplicity is motivated by the way…
We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots.
The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix…
We describe an efficient algorithm to compute a pseudotriangulation of a finite planar family of pairwise disjoint convex bodies presented by its chirotope. The design of the algorithm relies on a deepening of the theory of visibility…
We use microlocal sheaf theory to show that if two knots have Legendrian isotopic conormal tori, then the knots are isotopic or mirror images.
Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and…
We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the…