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Organizers |
Odd Khovanov homology, mutation, and branched double covers
by
Jonathan Bloom
Columbia University
We describe a thriftier construction of odd Khovanov homology, motivated by branched double covers. As a simple application, we prove that odd Khovanov homology is mutation invariant, and therefore that mod 2 Khovanov homology is mutation invariant. As a second application, we outline an extensive connection to monopole Floer homology.
Paper reference: arXiv:0903.3746, arXiv:0909.0816
Date received: October 6, 2009
A Generalization of the Turaev Cobracket and the Minimal Self-Intersection Number of a Curve on a Surface
by
Patricia Cahn
Dartmouth College
Goldman and Turaev constructed a Lie bialgebra structure on the vector space generated by free homotopy classes of loops on a surface. The Turaev cobracket gives a lower bound on the minimal number of self-intersection points of a loop in a given homotopy class. Chas proved that this lower bound is not sharp by providing examples of homotopy classes with zero cobracket that do not contain a simple representative. The Turaev cobracket factors through another operation μ, defined in the spirit of the Andersen-Mattes-Reshetikhin algebra. The operation μ also gives a lower bound on the minimal number of self-intersection points of a loop in a given homotopy class. We will show that this lower bound is sharp for homotopy classes α that do not contain a power of a curve. In particular, we show that the sum of the absolute values of the coefficients of μ(α) is always twice the minimal number of self-intersection points of a loop in α.
Date received: October 28, 2009
Categorification of Quandles: Strict 2-Quandles
by
Scott Carter
University of South Alabama
Coauthors: Alissa Crans, Mohamed Elhamdadi, Masahico Saito
A strict 2-quandle is a category in quandle. We show that the conjugation functor from groups to quandles extends to a function from strict 2-groups to strict 2-quandles. We also discuss weak notions of 2-quandles including coherent quandle objects in the category of categories in relation to braided monoidal 2-categories.
Date received: November 16, 2009
Topological Symmetry Groups of the Complete Graphs K4q+3
by
Dwayne Chambers
Claremont Graduate University
Coauthors: Professor Erica Flapan, John D. O'Brien
We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we show that only certain cyclic, dihedral, and products of two cyclic groups of odd order can occur as the topological symmetry group of an embedding of the complete graph K4q+3 in S3.
Date received: September 24, 2009
Linking numbers and the Conway polynomial of virtual links
by
Sergei Chmutov
Ohio State University, Mansfield
Coauthors: Z.Cheng, T.Dokos, J.Lindquist
Theorems of Hosokawa, Hartley, and Hoste state that the first non-trivial coefficient of the Conway polynomial is equal to a determinant of a certain matrix composed of the linking numbers between the components of the link. This determinant can be computed using the matrix-tree theorem from graph theory.
For virtual links there are two different types of the linking number and two Conway polynomials, ascending and descending. We generalize the theorem above to virtual links. In this case the determinant is related to the oriented version of the matrix-tree theorem. This is a joint work with my students Z.Cheng, T.Dokos, and J.Lindquist.
Date received: November 21, 2009
Crossing-free matchings in regular outerplane drawings
by
Paul C. Kainen
Department of Mathematics, Georgetown University
When does a regular graph have a circular layout with pages which are perfect matchings? It is shown that this holds for bipartite complete and hypercube graphs, as well as certain classes of 3-regular and 4-regular circulants. By a result from Overbay's thesis, any such graph must be bipartite, and a conjecture of Bernhart and the author holds that each bipartite graph does have such a "dispersible" embedding. Relations among book thickness type invariants will be considered.
Date received: November 30, 2009
Adventures in categorification
by
Mikhail Khovanov
Columbia University
This is an overview which starts with an old work of the author and moves on to exciting more recent constructions of Chuang-Rouquier, Lauda, Webster, and Zheng. We'll start with the induction and restriction functors between nilCoxeter algebras and show how they categorify the first Weyl algebra. Enlarging the algebra to the nilHecke algebra allows to categorify an integral version of the positive half of quantum sl(2). Forming cyclotomic quotients leads to categorification of irreducible representations of quantum sl(2). Finally, a distributed version of the cyclotomic quotients produces a categorification of tensor products of irreducible sl(2) representations.
Date received: November 30, 2009
Categorification of quantum groups
by
Mikhail Khovanov
Columbia University
Coauthors: Aaron Lauda
We'll report on a joint work with Aaron Lauda on categorification of positive halves of quantum universal enveloping algebras of simple Lie algebras.
Date received: November 30, 2009
Graphical calculus of Soergel bimodules in Khovanov-Rozansky link homology.
by
Daniel Krasner
Columbia University
Coauthors: Ben Elias
I will outline a graphical calculus of Soergel bimodules, developed by B. Elias and M. Khovanov, and describe how it can be used to construct an integral version of sl(n) and HOMLFYPT link homology, as well as prove functoriality of the latter.
Date received: November 15, 2009
A categorification of quantum sl(2).
by
Aaron Lauda
Columbia University
Crane and Frenkel conjectured that that the quantum enveloping algebra of sl(2) could be categorified at generic q using its canonical basis. In my talk I will describe a realization of this conjecture using a diagrammatic calculus.
Date received: November 30, 2009
Integrality of the Witten-Reshetikhin-Turaev 3-manifold invariant
by
Thang Le
Georgia Institute of Technology
Coauthors: K. Habiro
We give a new definition of the WRT invariant of integral homology 3-spheres using the Hopf link pairing and the twist element. This leads to a unified invariant (defined for all roots of unity) and integrality of the WRT invariant. The new definition might lead to a way to categorify the WRT invariant. This is joint work with K. Habiro.
Date received: November 24, 2009
Slicing Mixed Bing-Whitehead Doubles and Bordered Heegaard Floer Homology
by
Adam Levine
Columbia University
Using bordered Heegaard Floer homology, we give a wide class of links that are topologically but not smoothly slice: specifically, the all-positive Whitehead double of any iterated Bing double of any knot K with tau(K)>0. We also show that the all-positive Whitehead double of any link in the family of generalized Borromean rings is not smoothly slice; whether such links are topologically slice remains a major open question in the topology of four-manifolds.
Date received: November 23, 2009
Quantum Knots and Lattices
by
Samuel J. Lomonaco
University of Maryland Baltimore County (UMBC)
Coauthors: Louis K. Kauffman
Using the cubic honeycomb (cubic tessellation) of Euclidean 3-space, we 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. This quantum system, called a quantum knot system, is physically implementable in the same sense as Shor's quantum factoring algorithm is implementable.
To define a quantum knot system, we replace the standard three Reidemeister knot moves with an equivalent set of three moves, called respectively wiggle, wag, and tug, so named because they mimic how a dog might wag its tail. We argue that these moves are in fact more "physics friendly" because, unlike the Reidemeister moves, they respect the differential geometry of 3-space, and moreover they can be transformed into infinitesimal moves.
These three moves wiggle, wag, and tug generate a unitary group, called the lattice ambient group, which acts on the state space of the quantum system. The lattice ambient group represents all possible ways of moving a rope around in 3-space without cutting the rope, and without letting the rope pass through itself.
We then investigate those quantum observables of the quantum knot system which are knot invariants. We also study Hamiltonians associated with the generators of the lattice ambient group. We conclude with a list of open questions.
This talk is based on the following two papers:
http://arxiv.org/abs/0910.5891
http://arxiv.org/abs/0805.0339
Date received: December 1, 2009
Rack shadows and their invariants
by
Sam Nelson
Claremont McKenna College
Coauthors: Wesley Chang
A rack shadow is a set with an action by a rack, analogous to a vector space with a scalar multiplication. The set of colorings of a link diagram by a rack shadow yields various enhanced counting invariants. We will examine a few such invariants.
Date received: October 23, 2009
Cables of thin knots and bordered Heegaard Floer homology
by
Ina Petkova
Columbia University
We describe bordered Floer homology, defined by Lipshitz, Ozvath, and Thurston, in the case of torus boundary, and explain how to apply it to compute invariants of satellites of knots. In particular, we give a formula for HFK-hat of any (p, pn+1)-cable of a thin knot K in terms of the Alexander polynomial of K, tau(K), p, and n. We also give a formula for the Ozsvath-Szabo concordance invariant tau(K_{p, pn+1}) in terms of tau(K), p, and n.
Paper reference: http://arxiv.org/abs/0911.2679
Date received: November 16, 2009
The second quandle homology of Takasaki keis (quandles)
by
Jozef H. Przytycki
George Washington University
Coauthors: Maciej Niebrzydowski (UL at Lafayette)
Let G be an abelian group and T(G) its Takasaki quandle that is the quandle with g*h = 2h-g. We start the systematic study of the second homology of Takasaki quandles. We show, in particular, that H2(T(Z4k)) = Z22 ⊕Z2. We discuss the conjecture that for T(Zpk) the second homology is equal to Zpn(n-1)/2, for p an odd prime number.
Date received: December 4, 2009
Algebraic Structures Derived from Foams and TQFTs
by
Masahico Saito
University of South Florida
Coauthors: Scott Carter
Foams are surfaces with branch lines at which three sheets merge. The 2D TQFT of surfaces is characterized by means of Frobenius algebras, where saddle points correspond to multiplication and comultiplication. In this talk, we explore algebraic operations that branch lines derive under TQFT. In particular, we point out that Lie bracket and bialgebra structures can be found in infinitely many examples. Relations to the original Frobenius algebra structures are discussed both algebraically and diagrammatically. Foam skein modules of 3-manifolds are defined. Progresses made after the AMS Boca Raton meeting will be discussed.
Date received: November 19, 2009
Quandle homology and applications - An overview
by
Scott Carter and Masahico Saito
U. South Alabama and U. South Florida
This talk is an overview of quandles, their homology theories and applications, from history, basics, to most recent developments. We will start with basic constructions with some historical remarks, topological interpretations and the fundamental quandle, and variety of homology theories. Developments and most recent results in computational aspects and applications are discussed.
Date received: November 30, 2009
Circle homeomorphisms and shears
by
Dragomir Saric
The Graduate Center and Queens College of CUNY
The space of homeomorphisms Homeo(S1) of the unit circle S1 is a classical topological group which acts on S1. Homeo(S1) contains many important subgroups such as the infinite dimensional Lie group Diffeo(S1) of diffeomorphisms of S1, the group QS(S1) of quasisymmetric maps of S1, the characteristic topological group Symm(S1) of symmetric maps of S1, and many more. We use the shear coordinates on the Farey tesselation to parametrize the coadjoint orbit spaces M[(o)\ddot]b(S1)\Homeo(S1), M[(o)\ddot]b(S1)\QS(S1) and M[(o)\ddot]b(S1)\Symm(S1). To our best knowledge, this gives the only known explicit parametrization of the universal Teichmüller space T(H)=M[(o)\ddot]b(S1)\QS(S1).
Date received: November 25, 2009
SLarc algebra and what it categorifies
by
Radmila Sazdanovic
George Washington University
Coauthors: Mikhail Khovanov (Columbia University)
We introduce SLarc algebra and a categorification of the ring of one variable polynomials and show how this construction extends to the categorification of some classes of orthogonal polynomials.
Date received: December 6, 2009
Homologically Z2-thin knots have no 4-torsion in Khovanov homology.
by
Alexander Shumakovitch
GWU
I will show how to use the Bockstein spectral sequence to prove that homologically Z_2-thin knots have no 4-torsion in Khovanov homology. This completes the proof of the fact that the integer Khovanov homology of alternating knots is completely determined by their Jones polynomial and signature.
Date received: December 3, 2009
A colored sl(N)-homology for links in S^3
by
Hao Wu
George Washington University
I will explain how to generalize the Khovanov-Rozansky sl(N)-homology to a colored sl(N)-homology for links in S^3. I expect this colored homology to categorify the sl(N)-polynomial for links colored by wedge powers of the defining representation.
Date received: December 3, 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.