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The Einstein-Podolsky-Rosen paradox and quantum mysteries a la Mermin
by
Ali Eskandarian
The George Washington University
The Einstein-Podolsky-Rosen (EPR) paper of 1935 raised foundational questions about the completeness of quantum mechanics, which ultimately relate to the information content of the theory.
We review the progress made in addressing the EPR concerns through the works of Bell, Greenberger-Horne-Zeilinger (GHZ), L. Hardy, Mermin, and others that shed light on the nature of quantum states and the deficiencies of intuitive classical logic.
Date received: February 22, 2009
Configurations of Lagrangians in R^4
by
Ryan Hoban
University of Maryland
Abstract: The set of Lagrangian subspaces of a symplectic vector space is a submanifold of the Grassmannian that is invariant under the action of the symplectic group. In real dimension 4, this identifies with the three dimensional Einstein Universe. We will describe an invariant of quadruples of transverse Lagrangians which generalizes the classical cross ratio for quadruples of points in RP^1. This invariant can then be used to study the deformation space for some representations into the symplectic group.
Date received: February 26, 2009
Some remarks on the four-color problem
by
Paul C. Kainen
Department of Mathematics, Georgetown University
We discuss the implications of a result due to S. Gravier and C. Payan (Eur. J. Comb. 2002) regarding signed flip-paths for a new proof of the Four Color Theorem and its approximation in special cases.
Date received: February 27, 2009
Topological Quantum Information Theory
by
Louis H. Kauffman
University of Illinois at Chicago
Coauthors: Samuel J. Lomonaco Jr.
This abstract applies to a sequence of two talks. These talks will discuss the role of quantum topology and categorical structures related to quantum topology in the fields of quantum computing and quantum information theory. We will begin with a review of basic quantum information theory and the elements of diagrammatic and categorical approaches to quantum topology We then weave correspondences between these fields, discussing teleportation, topological gate construction, topological quantum computing and quantum knots.
Paper reference: quant-ph/0606114, arXiv:0804.4304
Date received: February 14, 2009
An Extended Bracket Polynomial for Virtual Knots
by
Louis H. Kauffman
University of Illinois at Chicago
This talk will discuss an extension of the bracket polynomial for oriented virtual knots and links that takes values in the module generated by isotopy classes of virtual 4-regular graphs over the ring of Laurent polynomials Q[A, A-1] where Q denotes the integers. This invariant is constructed by using an oriented state expansion and keeping as much combinatorial structure in the state sum as one can. Applications of the invariant and open problems will be discussed. A special case of this invariant is the arrow polynomial of the author and Heather Dye, a variant of the Miyazawa polynomial. We will discuss applications that involve the use of both the extended bracket and the arrow polynomial.
Paper reference: arXiv:0712.2546, arXiv:0810.3858
Date received: February 14, 2009
Graphs, links, and duality on surfaces
by
Slava Krushkal
University of Virginia
I will introduce a polynomial invariant of graphs on surfaces, generalizing the Tutte polynomial. This invariant satisfies a natural duality property, and it is closely related to the Bollobas-Riordan polynomial. I will also discuss a generalization of the Jones polynomial and Kauffman bracket for links in thickened surfaces, and their relation with the graph polynomial.
Date received: February 25, 2009
Quantum Knots and Lattices, or How Wiggle, Wag, and Tug Go Quantum
by
Samuel J. Lomonaco
University of Maryland Baltimore County
In the paper "Quantum Knots and Mosaics," QIP, (2008), the definition of quantum knots was based on the planar projections of knots (i.e., on knot diagrams) and the Reidemeister moves on these projections. In this paper, we take a different tack by creating a definition of quantum knots based on the cubic honeycomb decomposition of 3-space R³ (i.e., the cubic tessellation L_{ℓ} of R³ consisting of 2^{-ℓ}×2^{-ℓ}×2^{-ℓ} cubes) and a new set of knot moves, called wiggle, wag, and tug, which unlike the two dimensional Reidemeister moves are truly three dimensional moves. These new moves have been so named because they mimic how a dog might wag its tail.
We believe that these two different approaches to defining quantum knots are essentially equivalent, but that the above three dimensional moves have a definite advantage when it comes to the applications of knot theory to physics. More specifically, we contend that the new moves wiggle, wag, and tug are more "physics-friendly" than the Reidemeister ones. For unlike the Reidemeister moves, the new moves are three dimensional moves that respect the differential geometry of 3-space, which is indeed an essential component of physics. And moreover, unlike the Reidemeister moves, they can be transformed into infinitesimal moves and differential forms (which can in turn be integrated), which structures can be seamlessly interwoven with the equations of physics.
Our basic building block for constructing a quantum knot is a lattice knot, which is a knot in 3-space constructed from the edges of the cubic honeycomb L_{ℓ}. We then create a Hilbert space by identifying each edge of a bounded n×n×n region of the cubic honeycomb with a qubit. Lattice knots within this region then form the basis of a sub-Hilbert space K^{(ℓ,n)}. The states of K^{(ℓ,n)} are called quantum knots. The knot moves, wiggle, wag, and tug, are then naturally identified with the generators of a unitary group Λ_{ℓ,n}, called the lattice ambient group, acting on the Hilbert space K^{(ℓ,n)}.
This definition of a quantum knot can be viewed as a blueprint for the construction of an actual physical quantum system that represents the "quantum embodiment" of a closed knotted physical piece of rope. A quantum knot, as a state of this quantum knot system, represents the state of such a knotted closed piece of rope, i.e., the particular spacial configuration of the knot tied in the rope. The lattice ambient group Λ_{ℓ,n} represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement.
After defining quantum knot type, we investigate quantum observables which are invariants of quantum knot type. Moreover, we also study the Hamiltonians associated with the generators of the lattice ambient group.
This talk is based on the joint paper Quantum Knots and Lattices, or How Wiggle, Wag, and Tug Go Quantum by Lomonaco and Kauffman, (to appear on quant-ph.)
Date received: February 23, 2009
A Rosetta Stone for Quantum Computing
by
Samuel J. Lomonaco
University of Maryland Baltimore County
This talk will give an overview of quantum computing in an intuitive and conceptual fashion. No prior knowledge of quantum mechanics will be assumed.
The talk will begin with an introduction to the strange world of the quantum. Such concepts as quantum superposition, Heisenberg's uncertainty principle, the "collapse" of the wave function, and quantum entanglement (i.e., EPR pairs) are introduced. This part of the talk will also be interlaced with an introduction to Dirac notation, Hilbert spaces, unitary transformations, quantum measurement, and the density operator.
Simple examples will be given to explain and to illustrate such concepts as quantum measurement, quantum teleportation, quantum dense coding, and the first quantum algorithm, i.e., the Deutsch-Jozsa algorithm.
The PowerPoint slides for this talk will be posted.
Date received: February 23, 2009
SL_2(C)-Character Varieties of 2-bridge knots
by
Melissa Macasieb
University of Maryland
Coauthors: Ronald van Luijk
Kathleen Petersen
To every hyperbolic finite volume 3-manifold M, one can associate a pair of related algebraic varieties X(M) and Y(M), the SL_2(C)- and PSL_2(C)-character varieties of M. These varieties carry much topological information about M, but are in general difficult to compute. If M has one cusp, then both these varieties have dimension one. In this talk, I will also show how to obtain explicit equations for the character varieties associated to a family of hyperbolic 2-bridge knot complements.
Paper reference: arxiv:0902.2195
Date received: February 27, 2009
Projective knots
by
Rama Mishra
Indian Institute of Science Education and Research, Pune, India
Coauthors: Alan Durfee and Don Oshea
It is of interest to study knots in an arbitrary 3-manifold. Here we are interested in knots in the real projective 3-space RP3. As the unit circle S1 and the projective line RP1 are homeomorphic, a knot in RP3 may be regarded as a smooth embedding of RP1 in RP3. Embeddings of RP1 in RP3 are studied in algebraic geometry as a projective map of RP1 in RP3, which is a smooth embedding. We call them projective knots. We have seen that knots in S3 can be represented as a one point compactification of polynomial embeddings of R to R3 known as Polynomial knots. In this case we will see that knots in RP3 are represented by projective closure of embeddings given by rational functions from R to R3. In this talk we will discuss various questions related to projective knots.
Date received: February 19, 2009
Homology operations on homology of quandles
by
Maciej Niebrzydowski
University of Louisiana at Lafayette
Coauthors: Jozef H. Przytycki
In this joint work with Jozef Przytycki, we introduce the concept of the quandle partial differential equations and use solutions of these equations to construct homology operations on homology of racks and quandles.
Date received: February 13, 2009
Domination of knots and the Jones polynomial
by
Jozef H. Przytycki
George Washington University
Coauthors: Kouki Taniyama (Waseda and GWU)
We analyze the minimal degree of the Jones polynomial, degminVL(t), for links which have diagrams with a small number of negative crossings. In particular for a non-split nontrivial almost positive link link we show that degminVL(t) ≥ 1/2. This work is almost 19 years old but it is of new interest now due to the relation with Khovanov homology. We will describe the history of the problem (from 1987 for Przytycki, 1988 for Taniyama, and cooperation from 1990).
Date received: February 26, 2009
A Boolean equation related to the arc number of a chord diagram
by
Yongwu Rong
George Washington University
This talk is motivated by recent work by Kouki Taniyama, who defined arc numbers for chord diagrams and characterized those chord diagrams whose arc number is two. We consider the problem of finding an efficient algorithm to determine the arc number of a given chord diagram. We show that this problem can be reduced of the problem of finding the size of a minimal solution of a special kind of Boolean equation where no negation appears in its standard conjunctive normal form. While the later problem in general can be shown to be NP-hard, the computational complexity for computing the arc number may still be easier since the Boolean equations coming from chord diagrams may be rather special.
Date received: February 26, 2009
The mapping class group cannot be realized by homeomorphisms
by
Dragomir Saric
Queens College CUNY
Coauthors: Vladimir Markovic
The mapping class group MCG(S) of a closed surface S of genus at least 2 is quotient of the group of homeomorphisms Homeo(S) by the subgroup Homeo_0(S) of homeomorphisms homotopic to the identity. We show that MCG(S) does not homeomorphically lift to Homeo(S) which answers a question of Nielsen. This is a joint work with V. Markovic.
Paper reference: arXiv:0807.0182
Date received: February 23, 2009
Extended OC-TQFT
by
Simone Suarez
National University of Colombia campus Bogotá
Coauthors: Stella Huerfano
We will talk about Open Closed Topological Quantum Field Theories (OC-TQFT) that are a functors between the category of extended cobordisms and the ategory of knowledgeable Frobenius algebras. We will explain how the OC-TQFTs play a basic role in the extension of the Kovanov's Homology for links to a homology for tangles. We also want to compare the extension (of the Kovanov's homology or tangles) obtained by introducing a "extra" move to the ones used in Lauda's extension.
Date received: January 30, 2009
Circle immersions that can be divided into two arc embeddings
by
Kouki Taniyama
Waseda University and George Washington University
We give a complete characterization of a circle immersion that can be divided into two arc embeddings in terms of its chord diagram.
Paper reference: arXiv:0902.1478
Date received: February 11, 2009
Abstract graphs associated to link diagrams
by
Lorenzo Traldi
Lafayette College
Invariants derived from a knot or link diagram are usually defined by considering geometric properties of the diagram in the plane; a classical example is the checkerboard graph, which is defined using the diagram's complementary regions. There is another combinatorial structure associated to a diagram though, namely an abstract 2-in, 2-out directed graph, equipped with two partitions: its vertices are partitioned into those that correspond to positive crossings and those that correspond to negative crossings; and its edge-set is partitioned into the circuits that correspond naturally to the link components. Here "abstract" refers to a graph that is given without any preferred imbedding in the plane (or any other surface). The same construction applies to virtual link diagrams; the digraph corresponding to a virtual link diagram has vertices corresponding to the classical crossings.
There are several advantages to thinking of a diagram this way. One is that the combinatorial structure is simple: there are no complementary regions, cyclic orders of incident edges at vertices, under/overpassing arcs etc. Also, circuit partitions in 2-in, 2-out digraphs have received a great deal of attention from graph theorists in recent decades, so there are good combinatorial techniques available. Finally, abstract graphs represent classical and virtual knots and links with no necessary distinction between the two, just as abstract graph theory incorporates planar and non-planar graphs without distinguishing between them. For instance, abstract graphs have Reidemeister moves corresponding to the classical ones, but no Reidemeister moves corresponding to the virtual ones: they aren't needed because there is simply no information to which they might refer.
We have been able to verify that the Jones polynomial and Kauffman bracket can be derived from these abstract graphs. Identifying other invariants that can be derived from them is an open problem.
Paper reference: arXiv:0901.1451
Date received: January 27, 2009
sl(N)-homology for 1, 2-colored links
by
Hao Wu
George Washington University
Coauthors: Yasuyoshi Yonezawa
We generalize the Khovanov-Rozansky sl(N)-homology to links with components colored by 1 and 2. We expect that this homology categorifies the sl(N) quantum invariant of links colored by the defining representation and its 2-fold exterior power.
Date received: February 26, 2009
Copyright © 2009 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas.