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Organizers |
Towards a Classification of Cellular Automorphisms of Surfaces
by
Lowell Abrams
George Washington University
Coauthors: Daniel Slilaty, Wright State University
We describe an approach to constructing all cellular automorphisms of surfaces and their accompanying cellular structures, highlighting examples involving the torus.
Date received: May 5, 2010
A Generalization of the Turaev Cobracket for the Andersen-Mattes-Reshetikhin Algebra
by
Patricia Cahn
Dartmouth College
The Turaev cobracket Δ, defined on the free Z-module generated by free homotopy classes of loops on a surface, gives a lower bound on the number of self-intersection points of a loop α in a given free homotopy class. Turaev conjectured that Δ(α)=0 if and only if α is a power of a simple class. Chas constructed examples showing that this lower bound on the minimal self-intersection number is not an equality. These examples also disprove Turaev's conjecture. We construct a generalization μ of Δ, defined in the spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. We show that μ allows one to compute the minimal self-intersection number of a class α provided α is primitive (i.e., not a power of another class), and that an analogue of Turaev's conjecture holds for μ. In this talk, we will discuss the problem of computing the minimal self-intersection number of α when α is not primitive. We also extend μ to a subalgebra of the Andersen-Mattes-Reshetikhin algebra and explore its algebraic properties.
Paper reference: arXiv:1004.0532
Date received: April 16, 2010
Categorical Quandles and Knots
by
J Scott Carter
University of South Alabama
Coauthors: Alissa Crans, Mohamed Elhamdadi, Masahico Saito
In this talk I will sketch the definitions of a category in the category of groups and a category in the category of quandles. I will give examples of categories in the category of quandles. Two main examples will be given: one involves the tangent bundle of the three dimensional sphere, and the other is related to the Alexander module. Finally, I will sketch the definition of the fundamental 2-quandle of a classical knot.
Date received: May 12, 2010
(t, s) racks and their knot invariants
by
Jessica Ceniceros
Claremont McKenna College
Coauthors: Sam Nelson
Racks are self-distributive algebraic structures with axioms inspired by the Reidemeister moves. In this talk, we define an enhancement of the rack counting invariant for knots and links using the extra structure of (t-s) racks, a type of rack which generalizes Alexander quandles.
Date received: April 27, 2010
Orthogonal Quantum Group Invariants of Links
by
Qingtao Chen
University of Southern California
Coauthors: Lin Chen
The colored HOMFLY polynomial is a quantum invariant of oriented links in S associated with a collection of irreducible representations of each quantum group Uq(slN) for each component of the link. We will discuss in detail how to construct these polynomials and their general structure. Then we will discuss the new progress, Labastida-Marino-Ooguri-Vafa conjecture. The LMOV conjecture also gives the application of Lichorish-Millet type formula for links. The corresponding theory of colored Kauffman polynomial and orthogonal LMOV conjecture could also be developed in a similar fashion by using more complicated algebra structures. We prove several cases of this new conjecture. This is a joint work with Lin Chen.
Date received: May 12, 2010
Non-abelian theta functions a la Andre Weil
by
Razvan Gelca
Texas Tech University
Coauthors: Alejandro Uribe, University of Michigan
In 1964 Andre Weil pointed out to the existence of an action of a Heisenberg group on classical theta functions. This action induces, via a Stone-von Neumann theorem, the previously known Hermite-Jacobi action of the modular group. From the quantum mechanical point of view, theta functions, the finite Heisenberg group, and the Hermite-Jacobi action (given by discrete Fourier transforms) are the analogues, for a particle with periodic position and momenta, of what the square integrable functions, the Heisenberg with real entries, and the metaplectic representation (given by Fourier transforms) are for a free particle.
Non-abelian theta functions are defined as holomorphic sections of the Chern-Simons line bundle over the moduli space of connections on a surface. In the case where the gauge group is U(1) one recovers the classical theta functions. There is a combinatorial description of theta functions, as graphs colored by irreducible representations of quantum groups. The functions on the moduli space which are traces of holonomies along simple closed curves, the so called Wilson lines, admit a quantization as operators acting on theta functions, which can again be described using quantum groups. In the present talk we will explain that the algebra of quantum group quantizations of Wilson lines and the Reshetikhin-Turaev representation of the mapping class group of the surface are non-abelian analogues of the group algebra of the finite Heisenberg group and of the Hermite-Jacobi action.
Date received: March 11, 2010
Topological Quantum Computation and Topology in Quantum Computation
by
Tobias Hagge
U.T. Dallas
Topological quantum computing is a proposal to use exotic physical systems to compute link invariants and interpret them as quantum computations. We will provide a survey of some of the known connections between topological invariants, quantum computation, and quantum algorithms.
Date received: May 16, 2010
Bordered Heegaard Floer homology and the tau-invariant of cables
by
Jennifer Hom
University of Pennsylvania
We will use bordered Heegaard Floer homology to give a formula for the Ozsvath-Szabo concordance invariant tau of the (p, q)-cable of a knot K in terms of p, q, and two concordance invariants, tau(K) and epsilon(K), associated to the knot Floer complex of K. As a consequence, we will show that for any integer n, there exist knots K and K' with tau(K)=tau(K')=n such that tau of the (p, q)-cables of K and K' are not equal for any pair of relatively prime integers p and q. Finally, we will discuss some of the properties of epsilon; in particular, epsilon is strictly stronger than tau in determining obstructions to a knot being slice.
Date received: April 25, 2010
Quandles and multivariable polynomial invariants of twisted links
by
Naoko Kamada
Nagoya City University
Coauthors: Seiichi Kamada
In 2008 Bourgoin defined a twisted virtual link, or simply called a twisted link, which is a generalization of a virtual link. We introduce the quandle and a multivariable polynomial invariant of a twisted link. The latter is a generalization to twisted links of the multivariable polynomial invariants of virtual links defined independently by Y. Miyazawa, and H. A. Dye and L. H.Kauffman. We give lower bounds of real crossing numbers of twisted links.
Date received: April 3, 2010
On quandles and biquandles related to twisted virtual links
by
Seiichi Kamada
Hiroshima University
Coauthors: Naoko Kamada
We introduce two kinds of structures, called v-structures and t-structures, on biquandles. These structures are used for colorings of diagrams of virtual links and twisted virtual links such that the numbers of colorings are invariants. As an application, it is easily seen that Bourgoin's twofoil is not a virtual knot. The colorings can be used for quandle cocycle invariants of twisted virtual links. We also give a geometric interpretation of the group, in the sense of Bourgoin, and the quandle, in the sense of Naoko Kamada, of a twisted virtual link.
Date received: April 3, 2010
Integral TQFTs from Quantum Doubles, Gauge Equivariance, and Applications
by
Thomas Kerler
The Ohio State University
Coauthors: Qi Chen (WSSU)
We review TQFT constructions extending the Hennings invariant for 3-manifolds which start from a quasitriangular Hopf algebra H. We clarify how gauge equivalent Hopf algebras give rise to naturally isormophic TQFTs. We will also review past results in integral TQFTs in the traditional WRT approach and their relevance.
We will then show, more specifically, that if H=D(A) is the double of a Hopf algebra over a Dedekind domain R, and A is projective/free as an R-module we obtain a TQFT on puctured surfaces into the category of projective/free R-modules (with equivariant H-action). For closed surfaces projectivity survives.
We also show the double of the Borel algebra of quantum sl_2 is gauge equivalent to the product of quantum sl_2 itself and a cyclic group algebra. We infer the respective factorization of TQFT's and invariants. Combining our integrality result and the relation between sl_2 Hennings and WRT invariants established by Chen, Kuppum, and Srinivasan this yields an independent new proof of integrality of the WRT invariant.
Date received: May 11, 2010
Quantum Braids
by
Samuel J. Lomonaco
University of Maryland Baltimore County (UMBC)
Coauthors: Louis H. Kauffman
We will discuss quantum braids and their applications
Date received: May 14, 2010
Blackboard biracks and their link invariants
by
Sam Nelson
Claremont McKenna College
A blackboard birack is a set X with a solution B to the set-theoretic Yang-Baxter equation with invertibility properties making it suited for defining labeling invariants of blackboard-framed knots and links. We will see some examples of finite blackboard biracks and some of the knot and link invariants they define.
Date received: February 27, 2010
Quandle invariants of spatial graphs
by
Maciej Niebrzydowski
University of Louisiana at Lafayette
We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of long spatial graphs, we define an invariant based on the walks in the graph and quandle colorings.
Date received: May 9, 2010
Quantum Physics and Strings
by
William Parke
Thee George Washington University
The essence and strangeness of quantum physics is presented starting with the Feynman point of view. Some consequences in the nature flux strings, black holes, and computing are discussed.
Date received: May 14, 2010
The second quandle homology of the Takasaki quandle of an odd order abelian group
by
Jozef H. Przytycki
GWU and UTD
Coauthors: Maciej Niebrzydowski (ULL)
We prove that if G is an abelian group of odd order and T(G) its Takasaki quandle (that is a*b=2b-a) then there is an isomorphism from the second quandle homology H2Q(T(G)) to G ∧G where ∧ is the exterior product. In particular, for G=Zkn, k odd we have H2Q(T(Zkn)) = Zkn(n-1/2. Our proof is performed in five steps: First, we construct Cayley graph and Cayley 2-simplex of T(G). Then we choose a spanning tree for the Cayley graph and contract it. The result is the group Z(G×G) divided by relations [x, x]=0=[0, x] and [x, z]+[z, y] = [x, z-y+x]+[z-y+x, y]. Then we prove that for G generated by 2 elements the main result holds, in particular that [x, y] = -[y, x] . In the fourth step we show that the relation [z+x, z+y] = [z, y]+[x, z]+[x, y] holds and finally it leads to 2([w, z+y]-[w, z]-[w, y])=0 which for 2 not zero divisor, leads to linearity on the second component and from skew-symmetry to bilinearity and exterior product. Our result can be directly applied to classical knots as 2-(co)cycles give knot invariants. They can be also used to produce new nontrivial quandles.
Date received: March 26, 2010
A 2-category of dotted cobordisms and a universal odd link homology
by
Krzysztof Putyra
Columbia University
There are two kinds of Khovanov-type link homology theories: even introduced by M.Khovanov in 1999 and odd constructed by P.Ozsvath, Z.Szabo and J.Rasmussen in 2007. Both are derived from the cube of resolutions of a link diagram. However, the odd version is not given by a Frobenius algebra but only by a projective functor. In 2008, I rewrote the construction using chronological cobordisms, i.e. cobordisms equipped with special projections onto a unit interval (in fact it is a 2-category). On a side, I obtained a homology theory that specializes to both even and odd link homologies. Recently, I have found also a chronological version of dotted cobordisms and the neck-cutting relation, what simplifies the whole construction and gives in some sense a universal theory. Some implications are:
- non-existence of odd vesion of Lie theory
- there is only one dot in the odd theory over a field
Paper reference: arXiv:1004.0889
Date received: April 28, 2010
Computational complexity for topology and dynamics of boolean networks
by
Yongwu Rong
George Washington University
Coauthors: Rahul Simha, Guanyu Wang, Chen Zeng.
Networks have been of great interests in many areas in recent years. Such a network often consists of units with various levels of activities that evolve over time, mathematically represented by the dynamics of the network. The interaction between units is represented by the topology of a graph. An interesting problem is to study the connection between topology and dynamics of such networks. In particular, the so called reverse engineering problem asks for the topology of the network given information on its dynamics.
In this talk, we focus on a specific Boolean network model for biological networks. Under this model, the reverse engineering problem is naturally related to the Satisfiability Problem. We explain our results that (1) the decision problem can be solved in polynomial time, and (2) the problem of finding the minimal network solution is NP-hard.
Date received: May 15, 2010
Knot invariants from categorical fundamental quandles
by
Masahico Saito
University of South Florida
Coauthors: J. Scott Carter, Alissa Crans, Mohamed Elhamdadi
A strict 2-quandle is a category in which objects and morphisms are quandles and for which the structure maps are quandle morphisms. We define the fundamental strict 2-quandle topologically as certain homotopy classes of arcs in knot complements. Knot invariants motivated from these are defined diagrammatically, using arrows in knot diagrams. Applications are given for crossing numbers of virtual knots.
Date received: May 12, 2010
SLarc algebra, bimodules and functors
by
Radmila Sazdanovic
MSRI
Coauthors: Mikhail Khovanov
We describe various functors between categories of modules over SLarc algebra and explain which endomorphisms of the polynomial ring they categorify.
Date received: May 18, 2010
Comparing Invariants of 3-manifolds
by
Matt Sequin
The Ohio State University
Coauthors: Thomas Kerler (The Ohio State University)
The Kuperberg invariant and the Hennings invariant are both invariants of framed 3-manifolds constructed from Hopf algebras. It has been conjectured that the Kuperberg invariant associated to a Hopf algebra H is equal or closely related to the Hennings invariant constructed from D(H), the Drinfeld double of H. In this talk, we will first describe how to compute the lesser known Kuperberg invariant for non-involutory Hopf algebras. We will then discuss the well-studied Hennings invariant and describe the ways in which the two calculations are similar. Some preliminary results regarding the above conjecture will be given for Lens spaces and for specific Hopf algebras.
Date received: May 11, 2010
Color and orientation in sl(N)-homology
by
Hao Wu
George Washington University
An interesting fact I learned from Alex Shumakovitch is that Khovanov homology is not very sensitive to orientation changes. Roughly speaking, if you reverse the orientation of a component of a link, then the Khovanov homology changes only by an overall grading shift.
In this talk, I will explain why the colored sl(N)-homology is not very sensitive to orientation changes either. Roughly speaking, if you reverse the orientation of a component of a colored link and change its color from k to N-k, then the sl(N)-homology changes only by an overall grading shift. Since 2-1=1, this generalizes the above observation.
Date received: May 14, 2010
The orbits of Hurwitz action on systems of braids
by
Yoshiro Yaguchi
Hiroshima University
We study Hurwitz action of the n braid group Bn on the n-fold direct product of a braid group Bm, which can be used in study of braided surfaces, surface braids and orientable surface links. We determine the orbit of any n-tuple of the n distinct standard generators of Bn+1. In particular, the number of the elements of every orbit is (n+1)n-1. In addition, we show that any n-tuple of the n distinct standard generators of Bn+1 is transformed into any of those by Hurwitz action together with the action of Bn+1 by conjugation.
Date received: April 8, 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.