Related papers: Knotted fields and explicit fibrations for lemnisc…
Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on $T^3$. Some of the knot types considered in this article provide new examples of…
The present paper has a number of distinct purposes. First is to give a description of a class of electromagnetic knots from the perspective of foliation theory. Knotted solutions are then interpreted in terms of two codimension-2…
We introduce a new construction of surfaces in $D^2 \times B^2$, called knitted surfaces or BMW surfaces, which are described as the trace of deformations of knits. Here, knits are tangles obtained from classical braids from splicing at…
This paper develops a form of finite knot theory as a diagrammatic sequel to the ideal-stratum and deformation-persistence framework for knot types. Thick representatives in bounded ropelength sublevel spaces are studied through the finite…
This paper concerns twisted signature invariants of knots and 3-manifolds. In the fibered case, we reduce the computation of these invariants to the study of the intersection form and monodromy on the twisted homology of the fiber surface.…
We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for $n$-point functions. Perturbation theory leads us to…
We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We then describe the topology of the regular and singular fibres, in particular calculating their middle Betti numbers. For the…
Homomorphic expansions are combinatorial invariants of knotted objects, which are universal in the sense that all finite-type (Vassiliev) invariants factor through them. Homomorphic expansions are also important as bridging objects between…
Our aim of this and subsequent papers is to enlighten (a part of, presumably) arithmetic structures of knots. This paper introduces a notion of profinite knots which extends topological knots and shows its various basic properties.…
Using Hirasawa-Murasugi's classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman-Moore that L-space knots have no essential Conway spheres. In the…
We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the…
In this article we study Lefschetz fibration structures on knot surgery 4-manifolds obtained from an elliptic surface E(2) using Kanenobu knots $K$. As a result, we get an infinite family of simply connected mutually diffeomorphic…
Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai's theory of sutured manifold decomposition and contact topology. We implement this strategy for…
We define combinatorial invariants of Legendrian and transverse links in universally tight lens spaces using grid diagrams, generalizing [OST08] and prove that they are equivalent to the invariants defined in [BVVV13] and [LOSS09]. We use…
A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure we consider a knot…
A simple analytical way of creating superpositions of Bessel-Gaussian light beams with knotted nodal lines is proposed. It is based on the equivalence between the paraxial wave equation and the two-dimensional Schr\"odinger equation for a…
We present explicit algorithms for simplifying the topology of indefinite fibrations on 4-manifolds, which include broken Lefschetz fibrations and indefinite Morse 2-functions. The algorithms consist of sequences of moves, which modify…
A zigzag in a map (a $2$-cell embedding of a connected graph in a connected closed $2$-dimensional surface) is a cyclic sequence of edges satisfying the following conditions: 1) any two consecutive edges lie on the same face and have a…
Prime power fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots. In this paper, we give a new construction of non-slice knots that have the…
We study decomposition into simple arcs (i. e., arcs without self-intersections) for diagrams of knots and spatial graphs. In this paper, it is proved in particular that if no edge of a finite spatial graph $G$ is a knotted loop, then there…