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Organizers |
Arc index via alternating tangle decomposition
by
Yongju Bae
Kyungpook National University, Korea(University of South Alabama)
A diagram D of a link L can be decomposed as the sum of alternating tangles. In this talk we will show that an upper bound of the arc index α(L) can be obtained from the alternating tangle decomposition of D, indeed, the number of alternating tangles and the number of non-alternating edges.
Date received: November 5, 2010
A new state-sum formula for the Alexander polynomial
by
Samson Black
Simon Fraser University
We construct a diagram calculus to compute the Alexander polynomial of a braid, with an eye towards categorification.
Paper reference: arXiv:1002.4860
Date received: October 26, 2010
Width is not Additive
by
Ryan Blair
University of Pennsylvania
Coauthors: Maggy Tomova
In 1954, Schubert showed that bridge number of knots is additive with respect to connected sum. Gabai defined the related notion of width of a knot and used it in his proof of property R. We will present results that show width is not additive with respect to connected sum. This is joint work with Maggy Tomova.
Paper reference: arXiv:1005.1359
Date received: November 22, 2010
Finite type invariants obtained by counting surfaces
by
Michael Brandenbursky
Department of Mathematics, Vanderbilt University
Coauthors: Michael Polyak
A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain combinatorial types. These formulas generalize the calculation of a linking number by counting signs of crossings in a link diagram.
Until recently, explicit formulas of this type were known only for few invariants of low degrees. I will present simple formulas for an infinite family of invariants arising from the HOMFLY-PT polynomial. I will also discuss an interesting interpretation of these formulas in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram.
Date received: November 1, 2010
A Generalization of Turaev's Cobracket and the Minimal Self-Intersection Number of a Virtual String
by
Patricia Cahn
Dartmouth College
Turaev introduced a Lie cobracket on the free Z-module generated by nontrivial free homotopy classes of loops on a surface. Turaev's cobracket gives a lower bound on the minimum number of self-intersection points of a loop in a given free homotopy class. We introduce an operation μ which can be viewed as a generalization of Turaev's cobracket. We show that this operation gives an exact formula for the minimal number of self-intersection points of a loop in a given free homotopy class. Both Turaev's cobracket and μ can be extended to virtual strings, and both operations give a lower bound on the number of self-intersection points of a virtual string in a given virtual homotopy class. We show that the bound given by μ is similar to a bound given by Turaev's based matrix invariant, and is stronger than the bound given by Turaev's cobracket. We also show that μ gives an explicit formula for the minimal number of self-intersection points of a virtual string in certain virtual homotopy classes.
Paper reference: arXiv:1004.0532
Date received: November 17, 2010
An Introduction to Quandles, Their Homology, and Applications
by
J. Scott Carter
University of South Alabama
Abstract. The idea of a set with a binary operation that is self-distributive goes back to Takasaki (1942/43). In the early 1980s, Matveev and Joyce independently developed the axioms that we now call a quandle. They associated a quandle to the complement of a knot, and they showed that the knot quandle characterized the complement up to orientation reversing homeomorphism. Their construction is interesting to consider from the point of view of quandle 2-cocycles.
Quandle will be defined and exemplified with some nice geometric examples. Quandle (co)homology will be sketched, and some of the applications of quandle cocycle invariants that have been discovered by a variety of authors will be highlighted.
Date received: November 15, 2010
Two approaches to virtual Thistlethwaite's theorem
by
Sergei Chmutov
The Ohio State University, Mansfield
Coauthors: Clark Butler
There are two different generalizations of Thistlethwaite's theorem to virtual links. One is based on ribbon graphs and a topological version of the Tutte polynomial due to B.Bollobas and O.Riordan. Another involves a relative version of the Tutte polynomial of plane graphs found by Y.Diao and G.Hetyei. I explain a direct relation between the Bollobas-Riordan of ribbon graphs and relative Tutte polynomials plane graphs. This is a joint work with Clark Butler.
Paper reference: arXiv:1011.0072
Date received: November 23, 2010
The logic of quantum mechanics - take 2
by
Bob Coecke
Oxford, UK
It is now exactly 75 years ago that John von Neumann denounced his own Hilbert space formalism: ``I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.'' (sic) [1] His reason was that Hilbert space does not elucidate in any direct manner the key quantum behaviors. So what are these key quantum behaviors then? [2,3]
For Schrodinger this is the behavior of compound quantum systems, described by the tensor product [4, again 75 years ago]. While the quantum information endeavor is to a great extend the result of exploiting this important insight, the language of the field is still very much that of strings of complex numbers, which is akin to the strings of 0's and 1's in the early days of computer programming. If the manner in which we describe compound quantum systems captures so much of the essence of quantum theory, then it should be at the forefront of the presentation of the theory, and not preceded by continuum structure, field of complex numbers, vector space over the latter, etc, to only then pop up as some secondary construct.
Over the past couple of years we have played the following game: how much quantum phenomena can be derived from `compoundness + epsilon'. It turned out that epsilon can be taken to be `very little', surely not involving anything like continuum, fields, vector spaces, but merely a `two-dimensional space' of temporal composition (cf `and then') and compoundness (cf `while'), together with some very natural purely operational assertion, including one which in a constructive manner asserts entanglement; among many other things, trace structure (cf von Neumann above) then follow [5, survey]. In a very short time, this radically different approach has produced a universal graphical language for quantum theory which helped to resolve some open problems [6,7,8], and give a particularly elegant account on quantum classical interaction, on the basis of complementarity [9]. It also paved the way to automate quantum reasoning [10] and has even helped to solve problems outside physics, most notably in modeling meaning for natural languages [11].
This `categorical quantum mechanics' research program started with [12].
[1] M Redei (1997) Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Stud Hist Phil Mod Phys 27, 493-510.
[2] For von Neumann, initially these were the propositions that one could measure with certainty, an idea that he later abandoned in favor of the trace structure, which generates probability [1].
[3] Still, today for most physicists `quantum' is synonym for `Hilbert space', which of course is not unrelated to the dominant ``shut up and calculate''-conception of quantum theory.
[4] E Schroedinger, (1935) Discussion of probability relations between separated systems. Proc Camb Phil Soc 31, 555-563; (1936) 32, 446-451.
[5] B Coecke (2010) Quantum picturalism. Cont Phys 51, 59-83. arXiv:0908.1787
[6] B Coecke, B Edwards and RW Spekkens (2010) Phase groups and the origin of non-locality for qubits. ENTCS, to appear. arXiv:1003.5005
[7] R Duncan and S Perdrix (2010) Rewriting measurement-based quantum computations with generalised flow. ICALP'10.
[8] B Coecke and A Kissinger (2010) The compositional structure of multipartite quantum entanglement. ICALP'10. arXiv:1002.2540
[9] B Coecke and S Perdrix (2010) Environment and classical channels in categorical quantum mechanics. CSL'10. arXiv:1004.1598
[10] L Dixon, R Duncan & A Kissinger. dream.inf.ed.ac.uk/projects/quantomatic/
[11] B Coecke, S Clark & M Sadrzadeh (2010) Ling Anal 36. Mathematical foundations for a compositional distributional model of meaning. arXiv:1003.4394
[12] S Abramsky & B Coecke (2004) A categorical semantics of quantum protocols. LiCS '04. arXiv:0808.1023
Date received: November 8, 2010
Categorification of the Jones-Wenzl Projectors
by
Ben Cooper
University of Virginia
Coauthors: Slava Krushkal
We construct chain complexes P_n within Dror Bar-Natan's geometric category, whose graded Euler characteristic is the Jones-Wenzl projector p_n in the Temperley-Lieb algebra. These P_n are essentially unique and satisfy a graphical calculus up to homotopy. Consequences of our construction include families of knot invariants corresponding to higher representations of Uqsu(2) and a categorification of quantum spin networks.
Paper reference: arXiv:1005.5117
Date received: November 15, 2010
Evaluation of Fault-Tolerant Code Deformation
by
Andrew Cross
SAIC
Quantum error-correcting codes are an important tool for fighting noise in quantum computers. Topological quantum codes protect fragile quantum information by encoding it in the topology of a surface. A method called code deformation enables universal computation on the encoded data and involves changing that topology.
To reliably compute in the presence of noise, gate error rates need to be well below a constant accuracy threshold. High thresholds are desirable because they reduce accuracy requirements placed on quantum hardware. Raussendorf discovered that code deformation enables universal fault-tolerant quantum computation with thresholds near one percent [1]. The high threshold is achieved in a potentially realistic model where qubits interact with neighbors on a square two-dimensional grid.
Thresholds for this scheme are estimated by classically simulating quantum error correction [1, 2]. Thresholds for quantum computation are expected to be the same, but this has not been demonstrated. Our result is a conceptually simple method for fault-tolerantly deforming a planar code that retains a high accuracy threshold. The threshold for this method has been computed by direct simulation of code deformation within the stabilizer formalism.
This is joint work with Kevin Obenland. This work was supported by Science Applications International Corporation as internal research and development.
[1] Raussendorf and Harrington, Phys. Rev. Lett. 98, 190504 (2007).
[2] Fowler, Stephens, and Groszkowski, Phys. Rev. A 80, 052312 (2009).
Date received: November 29, 2010
Concordance in non-simply connected manifolds
by
Prudence Heck
Rice University
Call two knots in a 3-manifold M concordant if they co-bound a properly embedded annulus in M ×I. I will use L2-signature techniques to construct concordance invariants of null-homologous, homotopically essential knots in certain 3-manifolds. I will then construct an infinite family of non-concordant knots that are characteristic to some fixed knot.
Paper reference: arXiv:1005.4435
Date received: October 19, 2010
SO(3) Kauffman Homology of Graphs and Links
by
Matt Hogancamp
University of Virginia
Coauthors: Ben Cooper, Slava Krushkal
There is a well-known relationship between the SO(3) Kauffman polynomial for links, the chromatic polynomial for planar graphs, and the 2-colored Jones polynomial. In this talk I will describe a categorification of this relationship using a categorified Jones-Wenzl projector on two strands, living in Bar-Natan's category. Some elementary properties will be discussed, as well as future directions.
Date received: November 18, 2010
On some versions of Khovanov homology
by
Noboru Ito
Waseda University, Tokyo, Japan
In this talk, we introduce explicit chain homotopy maps and retractions for the invariance of Khovanov homology by Asaeda, Przytycki and Sikora using an interpretation of Khovanov homology by Viro. Viro gave another interpretation of Khovanov homology for generalized Jones polynomials of tangles introduced by Reshetikhin and Turaev. Then, if time permits, we try to apply this interpretation to a categorification of an operator invariant of colored tangles.
Date received: November 30, 2010
Khovanov-Rozansky Homology and Mutation
by
Thomas Jaeger
Michigan State University
We show that the reduced sln-homology defined by Khovanov and Rozansky is invariant under orientation and component preserving mutation when n is odd.
Date received: October 14, 2010
The A-Polynomial, Reidemeister Torsion and Quantum Invariants
by
Joanna Kania-Bartoszynska
National Science Foundation
Coauthors: Charles D. Frohman
Given a knot K in the 3-sphere denote by T the torus which is the boundary of the complement of K. The conjugacy classes of SU(2)-representations of the fundamental group of T are called the pillowcase. We use the Reidemeister torsion to construct a seminorm on the coordinate ring of the pillowcase whose radical is the A-ideal of the knot. A global formula for integrating against the Reidemeister torsion allows us to interpret it in terms of quantum invariants of the knot complement.
Date received: November 29, 2010
Categorifications and bilinear forms
by
Mikhail Khovanov
Columbia U
Bilinear forms help to make categorifications less of an art and more of a science.
Date received: November 29, 2010
Quantum Error Correction and Generalized Numerical Ranges
by
Chi-Kwong Li
College of William and Mary, George Washington University
Coauthors: Yiu-Tung Poon (Iowa State University) and Nung-Sing Sze (Polytechnic University of Hong Kong).
The numerical range is a useful tool for studying matrices and operators, and there are many generalizations of the concept motivated by theory and applications. In this talk, we focus on the higher rank numerical range, which arises from the study of quantum error correction. Recent results and open problems on the topic will be described.
Date received: November 4, 2010
Quantum Knots and Their Applications
by
Samuel Lomonaco
University of Maryland Baltimore County (UMBC)
Coauthors: Louis Kauffman
We begin by showing how to define a quantum system whose states, called quantum knots, represent a closed knotted piece of rope, i.e., represent the particular spatial configuration of a knot tied in a rope in 3-space. Such quantum systems, called quantum knot systems, are physically implementable in the same sense as Shor's quantum factoring algorithm is implementable.
Associated with a quantum knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving a rope in 3-space without cutting the rope, and without letting the rope pass through itself.
We then investigate those quantum observables of a quantum knot system which are knot invariants. We also investigate ways of associating Hamiltonians with the generators of the ambient group, and the resulting dynamic behavior of quantum knots as determined by SchroedingerÂ’s equation.
Paper reference: arXiv:0805.0339 & arXiv:0910.5891
Date received: November 28, 2010
A Turaev surface approach to Khovanov homology
by
Adam Lowrance
University of Iowa
Coauthors: Oliver Dasbach
We give an approach to Khovanov homology using a certain ribbon graph embedded on the Turaev surface of a link diagram. We discuss applications to adequate links and a relationship between Khovanov homology and the categorification of the Bollobas-Riordan-Tutte polynomial.
Date received: November 3, 2010
Rack modules and generalizations
by
Sam Nelson
Claremont McKenna College
The rack algebra is an associative algebra determined by a finite rack. Representations of the rack algebra, known as rack modules, can be used to enhance the rack counting invariant. In this talk we will preview three current projects involving generalizations of the rack algebra.
Date received: October 28, 2010
Homology theory in which distributivity replaces associativity
by
Jozef H. Przytycki
George Washington University
Homology theory of associative structures like groups and rings has been zealously studied throughout the past starting from the work of Hopf, Eilenberg, and Hochschild, but non-associative structures, like quandles, were neglected till recently.
Let *:X ×X → X be associative (that is X is a semigroup), then two classical homology
theories are group homology with
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The condition which can often replace associativity is distributivity and my talk will be devoted to homology of distributive structures with an eye on a hypothetical connection to Khovanov homology.
Consider a right self-distributive 2-argument operation *:X ×X → X (i.e. (a*b)*c = (a*c)*(b*c));
such a universal algebra is called a shelf.
Then the boundary operation ∂(*): RXn → RXn-1 given by
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One can approach our distributive homology from the more general perspective:
Let X be a set and let G(X) denote the set of all binary operations *:X×X → X on X. G(X) has a natural monoid (i.e. a semigroup with identity) structure with a composition *α*β given by a(*α*β)b=(a*αb)*βb, and the identity element operation *0 given by a*0b=a. We are mostly interested in submonoids of G(X) whose all pairs of elements are right distributive. that is for any pair of elements *α, and *β of the monoid, the operation *β is right distributive with respect to *α, ( (a*αb)*βc = (a*βc)*α(b*βc)). Such a monoid is called a monoid of shelf operations (resp. group of rack operations). The classical example of a shelf submonoid is generated by ∧ and ∨ operations of a distributive lattice on X.
For a given finite number of elements of a shelf submonoid *1, ..., *k we define a k-term derivative ∂(a1, ..., ak) = a1∂(1) + ... +ak∂(k) and study related homology and interrelations between various homologies.
If our operations satisfy idempotency condition x*x=x then (after Carter-Kamada-Saito) one can define k-term degenerate and quandle homology.
In the talk I will explore the above ideas and illustrate by two recent theorems about classical quandle homology (here G is an abelian group and the Takasaki quandle, T(G)=(G, *) is defined by a*b=2b-a.
Theorem 1. tor HnQ(T(Zp)) = Zpfn where fn is delayed Fibonacci sequence
fn = fn-1 + fn-3, and f(1)=f(2)=0, f(3)=1.
Theorem 2. HnQ(T(G) = G ∧G where ∧ is the exterior product, and G has an odd order.
Finally back to associativity; can one bring some of the above ideas back to classical objects? Yes, one can try out-associativity, that is (a*1b)*2c = a*2(b*1c) and linear combinations of various boundary operators.
Date received: November 28, 2010
Self duality of odd Khovanov homology
by
Krzysztof Putyra
Columbia University
Coauthors: Wojciech Lubawski, Polish Academy of Sciences
An odd Khovanov homology is a modification of sl_2 link homology defined in 2008 by P. Ozsvath and Z. Szabo. Instead of a symmetric algebra they used an antisymmetric one and got a different link homology theory that categorifies the Jones polynomial. One year later I generalized both constructions to a theory with three parameters. Together with W. Lubawski we were able to get rid of two of them. One of the consequences is a self-duality of odd homology conjectured by A. Shumakovitch.
Date received: November 6, 2010
Knot invariants from categorical quandles and Alexander modules
by
Masahico Saito
University of South Florida
Coauthors: J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi
A categorical quandle, called a strict 2-quandle, is a category and also is a quandle satisfying certain conditions. Definitions, constructions of examples, and the fundamental strict 2-quandle are reviewed. Colorings of knot diagrams by strict 2-quandles are defined, and a virtual knot invariant is defined in a manner similar to Alexander modules. Applications and variations of the invariant are given for virtual knots and spatial graphs.
Date received: November 28, 2010
Link invariants a la Alexander module
by
Oleg Viro
SUNY Stony Brook
Seifert's calculation of the Alexander module via Seifert surface can be modified to use with TQFT or Khovanov homology instead of the ordinary homology. With TQFT based on sl2, it gives rise to a construction assigning to a classical link a vector space with an operator whose trace is the value of the colored Jones polynomial at a root of unity. With Khovanov homology, it gives rise to a construction assigning to a surface in S3 ×S1 a bigraded module over the ring of Laurent polynomials.
Date received: November 24, 2010
Colored Morton-Franks-Williams inequalities
by
Hao Wu
George Washington University
We generalize the Morton-Franks-Williams inequality to the colored sl(N) link homology, which gives infinitely many new bounds for the braid index and the self linking number. A key ingredient of our proof is a composition product for the general MOY graph polynomial, which generalizes that of Wagner.
Date received: November 26, 2010
Unexpected local minima in the width complexes for knots
by
Alexander Zupan
The University of Iowa
In 1934, Goeritz exhibited a nontrivial diagram of the unknot that such any sequence of Reidemeister moves converting this diagram to the zero crossing diagram increases the number of crossings of the diagram. As an analogue, we produce a nontrivial embedding of the unknot such that any isotopy from this embedding to the thin position of the unknot increases knot width in the sense of Gabai. This resolves a question of Scharlemann, and we apply our result to demonstrate that the width complexes for knots developed by Schultens have infinitely many local minima that are not global minima.
Paper reference: arXiv:1008.5003
Date received: October 20, 2010
Copyright © 2010 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.