Local-moves on knots and products of knots II

We use the terms, knot product and local-move, as defined in the text of this paper. Let [Formula: see text] be an integer [Formula: see text]. Let [Formula: see text] be the set of simple spherical [Formula: see text]-knots in [Formula: see text]. Let [Formula: see text] be an integer [Formula: see text]. We prove that the map [Formula: see text] is bijective, where [Formula: see text]Hopf, and Hopf denotes the Hopf link. Let [Formula: see text] and [Formula: see text] be 1-links in [Formula: see text]. Suppose that [Formula: see text] is obtained from [Formula: see text] by a single pass-move, which is a local-move on 1-links. Let [Formula: see text] be a positive integer. Let [Formula: see text] denote the knot product [Formula: see text]. We prove the following: The [Formula: see text]-dimensional submanifold [Formula: see text] [Formula: see text] is obtained from [Formula: see text] by a single [Formula: see text]-pass-move, which is a local-move on [Formula: see text]-submanifolds contained in [Formula: see text]. See the body of this paper for the definitions of all local-moves in this abstract. We prove the following: Let [Formula: see text], and [Formula: see text] be positive integers. If the [Formula: see text] torus link is pass-move-equivalent to the [Formula: see text] torus link, then the Brieskorn manifolds, [Formula: see text] and [Formula: see text], are diffeomorphic as abstract manifolds. Let [Formula: see text] and [Formula: see text] be (not necessarily connected or spherical) 2-dimensional closed oriented submanifolds in [Formula: see text]. Suppose that [Formula: see text] is obtained from [Formula: see text] by a single ribbon-move, which is a local-move on 2-dimensional submanifolds contained in [Formula: see text]. Let [Formula: see text] be an integer [Formula: see text]. We prove the following: The [Formula: see text]-submanifold [Formula: see text] [Formula: see text] is obtained from [Formula: see text] by a single [Formula: see text]-pass-move, which is a local-move on [Formula: see text]-dimensional submanifolds contained in [Formula: see text].

Spatially Multiplexing of Metasurface for Manipulating the Focused Trefoil and Cinquefoil Vector Light Field

The trefoil and cinquefoil vector field are of essential significance for fundamental topology properties as the Hopf link and trefoil knots in the light field. The spatially multiplexing metasurfaces were designed with two sets of periodical nanoslits arranged alternately, each had independent geometric spiral phases and metalens phases to produce and focus vortex of the corresponding circular polarized (CP) light. By arranging the orientations of the two slit sets, the two CP vortices of the desired topological charges were obtained, the superposition of the vortices were realized to generate the vector field. With the topological charges of the vortices set to one and two, and three and two, respectively, the focused trefoil and cinquefoil vector light fields were acquired. The work would be important in broadening the applications of metasurface in areas as vector beam generations and topology of light field.

Legendrian Hopf Links

We completely classify Legendrian realizations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.

Geodesic fibrations for packing diabolic domains

We describe a theory of packing hyperboloid “diabolic” domains in bend-free textures of liquid crystals. The domains sew together continuously, providing a menagerie of bend-free textures akin to the packing of focal conic domains in smectic liquid crystals. We show how distinct domains may be related to each other by Lorentz transformations and that this process may lower the elastic energy of the system. We discuss a number of phases that may be formed as a result, including splay–twist analogues of blue phases. We also discuss how these diabolic domains may be subject to “superluminal boosts,” yielding defects analogous to shock waves. We explore the geometry of these textures, demonstrating their relation to Milnor fibrations of the Hopf link. Finally, we show how the theory of these domains is unified in four-dimensional space.

Triple-crossing number and moves on triple-crossing link diagrams

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of [Formula: see text]-crossing diagrams for every [Formula: see text] greater than one allows the definition of the [Formula: see text]-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.

2-Adjacency between some special links and pretzel links

This paper studies 2-adjacency between a 3-strand pretzel link and one of the Hopf link, the Solomon’s link and the Whitehead link by using the results that have been obtained about 2-adjacency between knots or links and their polynomials and etc. This paper shows that of all 3-strand pretzel links, only ordinary pretzel links are 2-adjacent to the Hopf link or the Solomon’s link or the Whitehead link. Conversely, these special links are not 2-adjacent to any other 3-strand pretzel links, except for themselves, respectively.