In mathematical knot theory, knots are defined as a smooth embedding of topological circle into R3 and all its images [1, 3, 4, 6, 7]. These knots have no open ends, such that they cannot be tied or untied. If a knot can deform (an ambient isopoty) into a geometric circle, then it belongs to a special subgroup of knot, called unknot. For a mathematical knot, if we cut at finite locations, then ...

A homogeneous knot is a generalization of alternating knots and positive knots. We determine the Rasmussen invariant of a homogeneous knot. This is a new class of knots such that the Rasmussen invariant is explicitly described in terms of its diagrams. As a corollary, we obtain some characterizations of a positive knot. In particular, we recover Baader’s theorem which states that a knot is posi...

In this paper we study kleinian groups of Schottky type whose limit set is a wild knot in the sense of Artin and Fox. We show that, if the “original knot” fibers over the circle then the wild knot Λ also fibers over the circle. As a consequence, the universal covering of S − Λ is R. We prove that the complement of a dynamically-defined fibered wild knot can not be a complete hyperbolic 3-manifold.

Knot modification of B-spline curves is extensively studied in the past few years. Altering one knot value, curve points move on well-defined paths, the limit of which can be computed if the knot value tends to infinity. Symmetric alteration of two knot values can also be studied in a similar way. The extension of these limit theorems for general synchronized modification of two knots is discus...

In this paper we investigate the computational complexity of knot theoretic problems and show upper and lower bounds for planarity problem of signed and unsigned knot diagrams represented by Gauss words. Due to the fact the number of crossing in knots is unbounded, the Gauss words of knot diagrams are strings over infinite (unbounded) alphabet. For establishing the lower and upper bounds on rec...

This paper describes the "scissor-knot-pusher," an instrument that greatly facilitates the execution of knot tying during laparoscopic operations. The instrument acts in essence as an extension of the surgeon's hand and, given its rigid structure, allows the surgeon full control of the process of knot tying. Additionally, after the knot has been tightened, it is possible to cut the suture witho...

In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the “quantum embodiment” of a closed knotted physical piece of rope. A quantum knot, as ...

The whitefly Bemisia tabaci is a major pest to agricultural crops. It transmits begomoviruses, such as Tomato yellow leaf curl virus (TYLCV), in a circular, persistent fashion. Transcriptome analyses revealed that B. tabaci knottin genes were responsive to various stresses. Upon ingestion of tomato begomoviruses, two of the four knottin genes were upregulated, knot-1 (with the highest expressio...

K not theory is usually understood to be the study of embeddings of topological spaces in other topological spaces. Classical knot theory, in particular, is concerned with the ways in which a circle or a disjoint union of circles can be embedded in R. Knots are usually described via knot diagrams, projections of the knot onto a plane with breaks at crossing points to indicate which strand passe...

Let K be a rationally null-homologous knot in a three-manifold Y . We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot K. As an application, we express the Heegaard Floer homology of rational surgeries on Y along a null-homologous knot K in terms of the filtered homotopy type of th...