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Organizers |
The Head and Tail of the Colored Jones Polynomial
by
Cody Armond
Louisiana State University
Coauthors: Oliver Dasbach
The colored Jones polynomial is a sequence of Laurent-polynomial knot invariants beginning with the Jones polynomial. We use techniques from skein theory to show that for alternating knots (and more generally, adequate knots) the j-th coefficient of the N-th colored Jones polynomial viewed as a sequence in N, will stabalize at N=j. For a knot with this property, we can define a pair of power series called the head and tail of the colored Jones polynomial.
In a joint work with Oliver Dasbach, we show many examples including torus knots and twist knots. Also, by considering the reduced A and B-graphs of an adequate diagram, we develop methods for calculating the head and tail of complicated knot from simpler knots. This allows us to determine the head and tail of a large family of knots, including, for example, all 2-bridge knots.
Date received: October 31, 2011
Cosmetic crossings and genus-one knots
by
Cheryl Balm
Michigan State University
Coauthors: Stefan Friedl, Efstratia Kalfagianni and Mark Powell
We will demonstrate some obstructions to the existence of cosmetic crossings in genus-one knots. As an application, we prove the nugatory crossing conjecture for several genus-one families of knots.
Date received: October 4, 2011
The Kakimizu complex of a link
by
Jessica Banks
University of Oxford
We give an introduction to the Kakimizu complex of a link, covering a number of recent results. In particular we will see that the Kakimizu complex of a knot may be locally infinite, that the Alexander polynomial of an alternating link carries information about its Seifert surfaces, and that the Kakimizu complex of a special alternating link is understood.
Date received: December 2, 2011
Graph homology and configuration spaces
by
Vladimir Baranovsky
University of California - Irvine
Coauthors: Radmila Sazdanovic
We will discuss the proof of the conjecture due to M. Khovanov relating the chromatic graph homology over algebra A defined by L. Helme-Guizon and Y. Rong, and the homology of a graph configuration space of a compact oriented manifold M which has cohomology algebra A. The latter construction was introduced by M. Eastwood, S. Huggett.
We show that there is a spectral sequence with the E_1 term given by the graph homology of A, and converging to the homology of the graph configuration space. When the graph is the complete graph on n vertices, the statement is essentially due to Bendersky and Gitler. In the same setting Totaro and others have proved that the spectral sequence degenerates if M is Kahler, or formal.
In their original work, Bendersky and Gitler have also conjectured that the differentials of the spectral sequence are, in some sense, given by the Massey products of M, and we prove this conjecture in the more general graph setting (earlier it was known for a complete graph on n=4 vertices, due to Felix and Thomas).
We will conclude by formulating two open questions related to the spectral sequence.
Date received: October 31, 2011
3-variable Laurent polynomial invariant of braids
by
Michael Brandenbursky
Vanderbilt University
We will construct and present a simple combinatorial formula for a 3-variable Laurent polynomial invariant I(a, z, t) of conjugacy classes in Artin braid group Bm on m strings. In addition we will show how the polynomial I(a, z, t) is derived from the HOMFLY-PT polynomial and that it satisfies the Conway skein relation in the variable z.
Date received: September 26, 2011
The Andersen-Mattes-Reshetikhin Bracket Counts Intersections
by
Patricia Cahn
Dartmouth College
Coauthors: Vladimir Chernov
Andersen, Mattes, and Reshetikhin defined a Poisson algebra structure on a quotient of the free module generated by homotopy classes of chord diagrams on an oriented surface. We show that the number of terms in the Poisson bracket of two free homotopy classes A and B is equal to the minimum number of intersection points of loops in the classes A and B when A and B are not equal. This generalizes work of Goldman and Chas. In particular, Chas showed that a similar statement does not hold for the Goldman bracket unless one of the classes A or B is simple.
Date received: November 26, 2011
Introducing knotted n-foams and how to construct invariants thereof
by
J. Scott Carter
University of South Alabama
Coauthors: Masahico Saito
The boundary of a surface foam is an trivalent graph. Analogously, a 3-foam can be defined that has a surface foam as its boundary. In this talk, I introduce the idea of a knotted n-foam and indicate how a homology theory of G-families of quandles is tailor-made to construct invariants of these knotted quantities. I will also give some interesting examples of 2-foams and indicate how to compute the cocycle invariants.
Date received: October 20, 2011
A Survey of Quandle Theory
by
Alissa S Crans
Loyola Marymount University
A quandle is a set equipped with two binary operation satisfying axioms that capture the essential properties of the operations of conjugation in a group and algebraically encode the three Reidemeister moves from classical knot theory. This notion dates back to the early 1980's when Joyce and Matveev independently introduced the notion of a quandle and associated it to the complement of a knot. We will focus on an introduction to the theory of quandles by considering examples, discussing quandle (co)homology and applications, and introducing recent work in this area.
Date received: November 24, 2011
On the Construction of Hamiltonian Operators for Adiabatic Quantum Computation
by
William De la Cruz
Cinvestav-IPN, Mexico city
Coauthors: Guillermo Morales Luna
Adiabatic Quantum Computation (AQC) has been applied to solve optimization problems. It is based on the construction of Hamiltonian operators which codify the optimal solution of the given optimization problem. AQC uses the Adiabatic Theorem to approximate solutions of the Schrodinger equation in which a slow evolution occurs.
The Hamiltonian operators used in AQC should be local for convenience which are expressed as sums of Hamiltonians operating over a subset number of qubits. We present a study on the construction of local Hamiltonian operators for graph problems whose instances belong to the graph classes expressible in monadic second order logic.
Date received: October 11, 2011
Galkin quandles and colorings of knots
by
Mohamed Elhamdadi
University of south Florida
Coauthors: E. Clark, X. Hou, M. Saito & T. Yeatman
Quandles appeared in many works on quasigroups (Stein 1957, Belousov 1960 and Galkin 1988). I will introduce the notion of Galkin quandles and show how we got in it through quasigroups. Each pointed abelian group gives a Galkin quandle. Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. Properties of Galkin quandles will be shown. We will demonstrate how Galkin quandles are used to color and then distinguish some knots.
Date received: November 24, 2011
Grid Movie Moves and Combinatorial Knot Floer Homology
by
Matthew Graham
Brandeis University
Recently, Sarkar showed that a smooth marked cobordism between two knots K1, K2 induces a map between the knot Floer homology groups of the two knots HFK(K1), HFK(K2). It has been conjectured that this map is well defined (with respect to smooth marked cobordisms). After outlining what needs to be shown to prove this conjecture, I will present my current progress towards showing this result for the combinatorial version of HFK. Specifically, I will present a generalization of Carter and Saito's movie move theorem to grid diagrams, give a very brief introduction to combinatorial knot Floer homology and then present a couple of the required chain homotopies needed for the proof of the conjecture.
Date received: November 14, 2011
The magic behind quantum computing: Chapter √-1
by
Jerzy Kocik
The soothingly graspable formalism of Quantum Mechanics (comprising of quite elementary concepts of linear algebra) contrasts strongly with profound interpretational problems of this formalism. Hence, not to discourage a reader, most expositions quickly move to the formalism and technical description of quantum algorithms, leaving a mathematician not trained in physics somewhat perplexed.
This gentler introduction to quantum computing honestly presents the strangeness of quantum nature of reality and is aimed to a non-physicist who ponders why quantum computers are possible.
Date received: November 24, 2011
Vassiliev Invariants of Virtual Framed and Virtual Legendrian Knots
by
Asa Levi
Dartmouth College
Coauthors: Patricia Cahn
We introduce virtual framed and virtual Legendrian knots. We then construct an isomorphism between the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots in the total space of the spherical cotangent bundle of a surface. This is philosophically related to the previous results of Fuchs-Tabachnikov, Goryunov, Hill and Chernov about isomorphisms of groups of Vassiliev invariants of (nonvirtual) Legendrian and topological knots.
Date received: November 11, 2011
Combinatorial Spanning Tree Models for Knot Homologies
by
Adam Levine
Brandeis University
Coauthors: John Baldwin
It is well-known that the Alexander and Jones polynomials of a knot can be computed as sums of monomials corresponding to spanning trees of the Tait graph of a diagram for the knot. In this talk, I describe recent progress on extending this approach to the knot homology theories that categorify these polynomials. Specifically, Baldwin and I have constructed a complex whose generators correspond to spanning trees, whose homology is isomorphic to the knot Floer homology of the knot, and whose differential can be described completely explicitly. Roberts used a similar approach to construct a spanning tree complex for Khovanov homology. The similarities between these two constructions suggest a possible strategy for relating the two theories.
Date received: November 25, 2011
2-strand twisting in Khovanov-Rozansky homology.
by
Andrew Lobb
Durham University
The twisting of two strands of a knot gives rise to a simple tensor factor of the knot homology. We consider maps of long exact sequences of homology groups which arise in this context, and give some results deduced from playing with these maps.
Date received: October 26, 2011
Quantizing Braids and Other Mathematical Structures
by
Samuel J. Lomonaco
University of Maryland Baltimore County (UMBC)
Coauthors: Louis H. Kauffman
We make explicit the general procedure for quantizing knots and braids, and show how this procedure can also be used to quantize other mathematical structures such as differential manifolds, Riemannian manifolds, and also algebraic structures.
We then discuss how these quantized objects can be used to create new quantum algorithms.
Date received: November 14, 2011
An odd categorification of the Tutte polynomial
by
Adam Lowrance
University of Iowa
Coauthors: Moshe Cohen
The Tutte polynomial is a graph and matroid polynomial which has a close relationship with the Jones polynomial. We construct a categorification of the Tutte polynomial for graphs and matroids that is modeled after the construction of odd Khovanov homology. We discuss an exact triangle and duality results for our categorification, as well as applications to alternating links.
Date received: October 25, 2011
Signs in totally twisted Khovanov homology
by
Andrew Manion
Princeton University
Recently, Lawrence Roberts constructed a 'totally twisted' variant of (delta-graded) Khovanov homology with coefficients modulo 2. His construction admits a spanning-tree model with an explicit differential. For knots, Jaeger showed that the totally twisted homology agrees with the standard Khovanov homology.
I will review the constructions of Roberts and Jaeger. Then I will describe how to add signs, resulting in a totally twisted theory over the integers which agrees with odd Khovanov homology for knots.
Date received: November 2, 2011
Ultrafast Gates for Trapped Ion Quantum Computing
by
Jonathan Mizrahi
University of Maryland, Joint Quantum Institute
Creating a functional quantum computer which can scale to many thousands of qubits is a very challenging task. The system must be isolated from the environment to an extreme degree, and it must be possible to exert exquisite control over the state of both the individual qubits and the collective ensemble of qubits. To date, a number of different physical systems have been proposed to deal with these challenges. One of the most promising is based on trapped atomic ions. I will talk about why ions make such good qubits, and how we manipulate them and entangle them. I will also talk about recent work we have done on creating ultrafast gates using mode-locked lasers.
Date received: November 17, 2011
Virtual Shadow Modules and their Link Invariants
by
Sam Nelson
Claremont McKenna College
Coauthors: Jackson Blankstein, Susan Kim, Catherine Lepel, Nicole Sanderson
Representations of the virtual shadow algebra and its twisted virtual analogue are used to enhance the counting invariant of virtual and twisted virtual knots. We will show that these enhanced invariants are sensitive to orientation and can distinguish knots not distinguished by the Miyazawa polynomial, the Arrow polynomial and the twisted Jones polynomial.
Date received: October 27, 2011
Mayer-Vietoris sequence for multispindles
by
Krzysztof Putyra
Columbia University
Coauthors: Jozef H. Przytycki
Homology theory for multispindles is motivated by rack homology. It is defined for any set X with a finite number of mutually and self distributive operations *1, ..., *n. Such a set is called a multishelf and, if each *i is idempotent, a multispindle.
Recently, Jozef H. Przytycki and the author have computed the homology for any finite distributive lattice, by reducing every lattice to a Boolean algebra with two elements B2. This method appeared to work for any 2-spindle (X, *1, *2) with absorption:
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In a general case, there is a long exact sequence of homology, similar to Mayer-Vietoris sequence known in Algebraic Topology. I will explain how this sequence arises for sets with distributive operations. As a result we will see that homology for spindles are almost determined by homology of quandles.
Date received: November 14, 2011
DNA Recombination Model via Spatial Graphs with 4-Rigid Vertices
by
Masahico Saito
University of South Florida
Coauthors: Angela Angeleska, Natasha Jonoska
We describe a model of RNA template guided recombination of DNA in certain kinds of ciliates. Genome rearrangement processes are modeled by 4-regular spacial graphs with rigid vertices, called assembly graphs. The rearranged DNA segments are modeled by certain types of paths in the assembly graphs called polygonal paths. The minimum number of such polygonal paths, and other aspects of assembly graphs are discussed. Applications of methods from knot theory are also discussed.
Date received: November 27, 2011
Computations of Szabo's geometric spectral sequence in Khovanov homology
by
Cotton Seed
Princeton University
Szabo has introduced a new geometric spectral sequence in Khovanov homology.
I will present results of computations of this spectral sequence and a number of conjectures concerning its structure. I will present some partial results towards these conjectures, in particular, I will show that the two ways of defining the reduced theory agree.
Finally, I will discuss the techniques used to compute the spectral sequence and, if time permits, sketch some ongoing work to relate this spectral sequence to the spectral sequence from Khovanov homology to the Heegaard Floer homology of the double-branched cover defined by Ozsvath-Szabo.
Date received: November 7, 2011
Grid2Mosaic2Grid: A Complete Pair of Polynomial Knot Algorithms
by
Omar Shehab
University of Maryland, Baltimore County
Coauthors: Sumeetkumar Bagde (University of Maryland, Baltimore County)
Samuel J. Lomonaco Jr. (University of Maryland, Baltimore County)
Tame Knot Diagrams can be represented by two different discrete structures, namely, Grid Diagrams and Knot Mosaics. This report proposes two polynomial time algorithms for translations between Grid Diagrams and Knot Mosaics. It is shown that that the time complexity of both algorithms is O(n3). These results prove that Grid Diagrams and Knot Mosaics are topologically equivalent. This equivalence is efficiently computable. We also conjecture that the two Cromwell moves of Grid Diagrams, i.e. Castling and Stabilization, are equivalent to sequences of planar moves defined for Knot Mosaics. These equivalences are also conjectured to be polynomially computable.
Date received: November 23, 2011
An alternative approach to hyperbolic structures on link complements
by
Anastasiia Tsvietkova
University of Tennessee, Knoxville
Coauthors: Morwen Thistlethwaite
Thurston demonstrated that every link in S3 is a torus link, a satellite link or a hyperbolic link and these three categories are mutually exclusive. It also follows from work of Menasco that an alternating link represented by a prime diagram is either hyperbolic or a (2, n)-torus link.
A new method for computing the hyperbolic structure of the complement of a hyperbolic link, based on ideal polygons bounding the regions of a diagram of the link rather than decomposition of the complement into ideal tetrahedra, was suggested by M. Thistlethwaite. Although the method is applicable to all diagrams of hyperbolic links under a few mild restrictions, it works particularly well for alternating (non-torus) links. The talk will introduce the basics of the method. Some applications will be discussed, including a surprising rigidity property of certain tangles, a new numerical invariant for tangles, and formulas that allow one to calculate the volume of 2-bridged links directly from the diagram.
Date received: November 5, 2011
Khovanov's diagram algebras and bordered Floer homology
by
Stephan Wehrli
Syracuse University
Coauthors: Denis Auroux, J. Elisenda Grigsby
I will discuss a connection between certain Khovanov- and Heegaard Floer-type invariants for knots, braids, and 3-manifolds. Specifically, I will explain how the 1-strand part of the bordered Floer bimodule associated to the branched double-cover of a braid is related to a similar bimodule defined by Khovanov and Seidel.
I will further present a partial result relating the k-strand part of the bordered Floer algebra to one of the cellular algebras studied by Chen-Khovanov and Brundan-Stroppel.
Date received: November 18, 2011
The sl(N) Rasmussen invariants of (2, 2k+1) cable knots
by
Hao Wu
George Washington University
We study the sl(N) Rasmussen invariants of (2,2k+1) cable knots using simplified Khovanov-Rozansky chain complexes of 2-braids.
This work was motivated by recent results of Lobb.
Date received: November 26, 2011
The Skein Algebra of Punctured Surfaces
by
Tian Yang
Mathematics Department, Rutgers University-New Brunswick.
Coauthors: Julien Roger
The Kauffman bracket skein module K(M) of a 3-manifold M is defined by Przytycki as an invariant for framed links in M satisfying the Kauffman skein relation. For a compact oriented surface S, it is shown by Bullock, Przytycki, Sikora, Frohman and Kania-Bartoszynska that K(S×I) is a quantization of the SL2C-characters of the fundamental group of S in the Goldman-Weil-Petersson Poisson structure.
In a joint work with J.Roger, we define a skein algebra of a punctured surface as an invariant for not only framed links but also framed arcs in S×I satisfying the skein relations of crossings both in the surface and at punctures. This algebra quantizes a Poisson algebra of loops and arcs on S in the sense of deformation of Poisson structures. This construction provides a tool to quantize the decorated Teichmüller space; and the key ingredient in this construction is a collection of geodesic lengths identities in hyperbolic geometry which generalizes/is inspired by Penner's Ptolemy relation, the trace identity and Wolpert's cosine formula.
Date received: October 3, 2011
Milnor invariants and the HOMFLYPT polynomial
by
Akira Yasuhara
Tokyo Gakugei University
Coauthors: Jean-Baptiste Meilhan (Univ of Grenoble I)
We give formulas expressing Milnor's isotopy invariants of a link L in the 3-sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor invariants of L vanish for all sequences with length at most k, then any Milnor invariant of L with length between 3 and 2k+1 can be represented as a combination of HOMFLYPT polynomial of knots obtained from the link by certain band sum
Date received: October 22, 2011
Pants distance, twist number, and volume of hyperbolic 2-bridge knots
by
Alexander Zupan
University of Iowa
We adapt an approach of Bachman and Schleimer to define the pants distance of a bridge splitting for a knot K in a 3-manifold M. If K is a hyperbolic 2-bridge knot in the 3-sphere, the pants distance of a 2-bridge decomposition of K is closely related to the twist number and the volume of K. We will discuss evidence in support of a more general relationship between pants distance and hyperbolic volume for other families of knots.
Date received: November 1, 2011
Copyright © 2011 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.