Related papers: Crowell's state space is connected
For an arbitrary positive integer $n$ and a pair $(p, q)$ of coprime integers, consider $n$ copies of a torus $(p,q)$ knot placed parallel to each other on the surface of the corresponding auxiliary torus: we call this assembly a torus…
New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used…
In this paper the geometry of two and three-qubit states under local unitary groups is discussed. We first review the one qubit geometry and its relation with Riemannian sphere under the action of group $SU(2)$. We show that the…
We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a…
Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. According to the conjecture, certain boundary slopes are detected by the sequence of degrees of the colored Jones polynomials. We…
We study the relation between perturbative knot invariants and the free energies defined by topological string theory on the character variety of the knot. Such a correspondence between SL(2;C) Chern-Simons gauge theory and the topological…
Let $D$ be a diagram of an alternating knot with unknotting number one. The branched double cover of $S^3$ branched over $D$ is an L-space obtained by half integral surgery on a knot $K_D$. We denote the set of all such knots $K_D$ by…
We give an elementary, self-contained proof of the theorem, proven independently in 1958-9 by Crowell and Murasugi, that the genus of an alternating knot equals half the breadth of its Alexander polynomial, and that applying Seifert's…
We explicitly construct an Archimedean order unit space whose state space is affinely isomorphic to the set of quantum commuting correlations. Our construction only requires fundamental techniques from the theory of order unit spaces and…
Ozsvath and Szabo proved that knot Floer homology determines the genera of knots in S^3. We will generalize this deep result to links in homology 3-spheres, by adapting their method. Our proof relies on a result of Gabai and some…
We produce a facial state sum on plane diagrams of a knot or a link which admits an invariant specialization under Polyak's recent set of generating of 4 Reidemeister moves. Thus an isotopy invariant of framed links is obtained. Each state…
We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements…
This paper is expository and is accessible to students. We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible…
The time evolution of even and odd squeezed states, as well as that of squeezed number states, has been given in simple, analytic form. This follows experimental work on trapped ions which has demonstrated even and odd coherent states,…
An intrinsic relation between maximally entangled states and entanglement measures is revealed, which plays a role in establishing connections for different entanglement quantifiers. We exploit the basic idea and propose a framework to…
We derive new existence results for tight contact structures on certain 3-manifolds which can be presented as surgery along specific knots in S^3. Indeed, we extend our earlier results on knots with maximal Thurston-Bennequin number being…
We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We…
We show that the length of a qubit-qutrit separable state is equal to the max(r,s), where r is the rank of the state and s is the rank of its partial transpose. We refer to the ordered pair (r,s) as the birank of this state. We also…
New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear…
In this paper we provide a computer assisted proof that about two thousand surgeries far away from the ideal point in the hyperbolic Dehn filling space of the figure-eight knot complement are infinitesimally projectively rigid. We also…