Related papers: State Surfaces of Links
We generalize a theorem of Finkelstein and Moriah and show that if a link $L$ has a $2n$-plat projection satisfying certain conditions, then its complement contains some closed essential surfaces. In most cases these surfaces remain…
Soft interfaces can mediate interactions between particles bound to them. The force transmitted through the surface geometry on a particle may be expressed as a closed line integral of the surface stress tensor around that particle. This…
Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement…
The fundamental role of on-shell diagrams in quantum field theory has been recently recognized. On-shell diagrams, or equivalently bipartite graphs, provide a natural bridge connecting gauge theory to powerful mathematical structures such…
A state generating is introduced to determine the Jones polynomial of a link. Formulae for two infinite families of knots are shown by applying this method, the second family of which are proved to be non-alternating. Moreover, the method…
In this paper, we study some relationships existing between some particular mathematical structures: discrete surfaces coming from discrete topology and mathematical morphology, poset-based connected manifolds coming from discrete topology,…
We introduce and analyze numerical companion matrix methods for the reconstruction of hypersurfaces with crossings from smooth interpolants given unordered or, without loss of generality, value-sorted data. The problem is motivated by the…
A classification of spanning surfaces for alternating links is provided up to genus, orientability, and a new invariant that we call aggregate slope. That is, given an alternating link, we determine all possible combinations of genus,…
In this paper, we formulate a construction of ideal coset invariants for surface-links in $4$-space using invariants for knots and links in $3$-space. We apply the construction to the Kauffman bracket polynomial invariant and obtain an…
This article reports on a program to obtain and understand coherent states for general systems. Most recently this has included supersymmetric systems. A byproduct of this work has been studies of squeezed and supersqueezed states. To…
We study surface knots in 4-space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the…
It is argued that spectral features of quantal systems with random interactions can be given a geometric interpretation. This conjecture is investigated in the context of two simple models: a system of randomly interacting d bosons and one…
In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon-Litherland bilinear forms associated to essential state surfaces of the 2-bridge knots. We show that the…
This paper introduces a new algebra, the crossing algebra, that is applied to count the number of components for arborescent knots, links, tangles or states (of a state polynomial expansion such as the Kauffman bracket). This algebra is…
Laplacian matrices of weighted graphs in surfaces $S$ are used to define module and polynomial invariants of $Z/2$-homologically trivial links in $S \times [0,1]$. Information about virtual genus is obtained.
We analyze the surface tension exerted at the interface between an active fluid and a solid boundary in terms of tangential forces. Focusing on active systems known to possess an equation of state for the pressure, we show that interfacial…
In this paper, we construct mock Alexander polynomials for starred links and linkoids in surfaces. These polynomials are defined as specific sums over states of link or linkoid diagrams that satisfy $f=n$, where $f$ denotes the number of…
A system of strongly interacting fermions in a solid state is discussed. A structure of singlet and triplet coupled 2-particle states and their excitation spectra are investigated. It is shown that an account of intersite fermion…
Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain…
This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic, combinatorial and topological structures. We include a description of previous work on the construction of Hilbert…