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Organizers |
Branched Spines approach to combinatorial Heegaard Floer homology
by
Carl Hammarsten
George Washington University
A 3-dimensional closed manifold Y represented by its branched spine has a canonical Heegaard decomposition. We present this decomposition graphically in the form of a Strip Diagram. We then show that strip diagrams have nice properties which greatly simplify the calculation of Heegaard Floer homology. Motivated by this work, we present a combinatorial definition of a chain complex which we expect to be homotopically equivalent to the Heegaard Floer one, yet significantly smaller. Finally, we consider the presentation of a branched spine by its O-graph and show how to reformulate our definition in these terms.
Date received: May 7, 2014
State Models and the Jones Polynomial
by
Louis H. Kauffman
University of Illinois at Chicago
This talk will begin with a recap of the construction of the (Kauffman) bracket state
summation for the Jones polynomial and its relationship with the diagrammatic
Temperley-Lieb algebra and hence with the original representation of the braid group
to that algebra by Vaughan Jones. We then discuss how the bracket state sum is a
special case of the formalism for the partition function
for the Potts model in statistical mechanics. We show how the bracket state sum
fits into the framework of other invariants and into the framework of the
Yang-Baxter equation.We then keep looking at the idea of combinatorial state sums
with forays into the Alexander polynomial, the arrow polynomial in
virtual knot theory and formulations of Khovanov
homology that are closely related to state summations. This talk has a semi-historical form,
but it is intended to explore the possibilities in these constructions that we understand only
a little bit at the present time.
Date received: May 5, 2014
Virtual Knot Cobordism
by
Louis H. Kauffman
Math UIC, Chicago, IL 60607-7045
This talk will outline, with examples, a theory of virtual knot cobordism that generalizes classical knot
cobordism and has many interesting properties and problems. One particular problem we are
interested in is the classification of virtual knots up to band-passing (as in the classical case this means switching
pairs of oppositely oriented arcs) and the relationship of this classification with concordance of virtual knots.
Date received: May 6, 2014
From Ehrfenfest hat to quantum computing
by
Jerzy Kocik
Southern Illinois University
Krawtchouk matrices bring a rather intriguing web of connections between the classical random walk, Hadamard-Sylvester operators, and representations of groups su(2) and sl(2). The aspect that connects them to quantum computing will be presented. This is a “quantum conclusion” to the previous talk on Krawtchouk matrices.
Date received: May 5, 2014
The Paradoxical IT
by
Samuel Lomonaco
University of Maryland Baltimore County (UMBC)
Since early antiquity, philosophers have spoken of IT (i.e., the physical world), or NOT IT. Einstein insisted on a DEFINITE IT with his Principle of Reality and Principle of Locality. In this talk, we illustrate the paradoxical IT with the Greenberger-Horne-Zeilinger (GHZ) paradox. We will begin by first describing a physical device, and then prove that it cannot be built. Next we show how to build this very same device using the laws of quantum mechanics. Is IT ambiguous?
Date received: May 8, 2014
Knot groups generated by two conjugate elements
by
Melissa Macasieb
University of Maryland
Coauthors: Michel Boileau
We consider an infi nite family of groups generated by two conjugate elements and show that such groups cannot be isomorphic to any known knot group. This family of groups is closely related to the well-known and well-understood family of 2-bridge knot groups.
Date received: May 4, 2014
Non-orthodox approaches in quantum computing
by
Robert Owczarek
University of New Mexico
Some less orthodox approaches to (some aspects) of quantum computing
will be presented: the potential role of dynamic symmetry of quantum groups in fighting
errors; the knotted vortex structures as a means for quantum computing;
graphene, synthetic metals and other materials with Dirac particles
as charge carriers as materials of choice for quantum computing
Date received: May 5, 2014
q-polynomial invariant of rooted trees
by
Jozef H. Przytycki
George Washington University and University of Maryland
It seems to be appropriate to propose, on this 30 anniversary of the Jones polynomial, yet another new, to my best knowledge, polynomial. It is a very simple polynomial of rooted trees but intimately connected to the Jones polynomial via the Kauffman bracket polynomial and the Temperley-Lieb algebra. Thus for a rooted tree (T, v) we construct the polynomial Q(T, v) ∈ Z[q]. Here is the definition, starting from plane rooted tree (later proved to be independent on plane embedding).
Definition: For a plane tree T with a root v we define a polynomial Q(T) by a recursive relation: Q(T) is equal to the sum over all leaves, vi, of T of the polynomials Q(T-vi) with the coefficients qr(vi), where r(vi) is the number of edges of T to the right of vi. We normalize Q(T) to be 1 at the root. Among many possible ways to define planar rooted tree polynomial, Q(T) has an advantage of being independent of plane embedding so it is rooted tree invariant. We give a formula for a root change and for a polynomial of a root product T=T1∨T2. We get: Q(T1 ∨T2) = ((E(T1)+E(T2)) || (E(T1)))qQ(T1)(Q(T2) Where (a || b)q denotes a Gauss polynomial, that is q-binomial coefficient. For the use in knot theory we consider plane rooted tree with delay function fT: E(T) → Z, where Q(T, fT) is the sum only over those leaves with value function no more than 1, and fT-v=fT -1.
The application of Q(T) to a lattice crossing will be presented in a join paper with Mieczyslaw Dabkowski and Changsong Li.
In the broader context we will show (after Loday) that plane rooted trees form an almost simplicial set and we speculate about changing presimplicial modules into q-polynomials, by considering ∑i=0nqidi in place of ∑i=0n(-1)idi.
Date received: April 3, 2014
Odd homological operations on Khovanov homology
by
Alexander Shumakovitch
The George Washington University
Coauthors: Krzysztof Putyra
We discuss homological operations between even and odd Khovanov homology theories. These operations are defined via the unified even/odd Khovanov homology theory developed by Putyra and give rise to new knot invariants with interesting properties.
Date received: May 8, 2014
Introduction to Khovanov type graph homology for non-commutative algebras
by
Jing Wang
George Washington University
Coauthors: Jozef H. Przytycki
Few years after Khovanov homology was introduced as the categorification of Jones polynomial for knots, its version for graphs was developed by Helme-Guizon and Rong, Later Przytycki observed the relation with Hochschild homology.
In this talk, I will introduce a more general definition of this Khovanov type graph homology for non-commutative algebras. In particular, we use the language of homology of a small category with functor coefficients and consider directed graphs with the idea of multi-paths proposed by Turner and Wagner.
Date received: May 7, 2014
Random Walk Invariants of String Links via Representation Theory
by
Yilong Wang
The Ohio State University
Coauthors: Thomas Kerler, The Ohio State University
In this talk, we will introduce two representations of the monoid of string links, one coming from the random walk model proposed by X.S. Lin et al, the other arising as the quotient between the restriction of a graded U-1(sl2) representation to its degree 1 component and that to its degree 0 component. We will show briefly how both of the representations can be viewed as generalizations of the unreduced Burau representations on the braid group. Then we will state our main result claiming that these two representations are actually isomorphic, and if time permits, we will give a hint on the proof of the theorem.
Date received: May 6, 2014
Commuting actions of sl(2) and S_n on sutured annular Khovanov homology
by
Stephan Wehrli
Syracuse University
Coauthors: Eli Grigsby and Tony Licata
To a link L in a thickened annulus, Asaeda-Przytycki-Sikora assigned a Khovanov-type homology theory which categorifies the skein module of the thickened annulus and which is related to a certain knot Floer homology by work of Roberts. In this talk, I will show that this homology theory carries a natural action of sl(2) and, in the case where L is the n-cable of a framed knot K, a commuting action of the symmetric group S_n. In the case where K is the 0-framed unknot, we recover classical Schur-Weyl duality for the nth tensor power of the fundamental representation of sl(2). This is joint work with Eli Grigsby and Tony Licata.
Date received: May 2, 2014
Two formulas for the MOY graph polynomial and an explicit so(6) homology
by
Hao Wu
George Washington University
I will explain the generalizations of Jaeger's Formula and Composition Product to the MOY graph polynomial. These generalized formulas implies that the 2-colored sl(4) polynomial is equal to the so(6) Kauffman polynomial. Thus, the 2-colored sl(4) homology categorifies the so(6) Kauffman polynomial.
Date received: May 9, 2014
Twist Spinning of Knots and Graphs
by
Seung Yeop Yang
George Washington University
Coauthors: Jozef H. Przytycki
We start by defining basic spinning of a classical knot to a knotted sphere, defined by E. Artin in 1925. We describe a work of Zeeman and Epstein, 1960, and construct a twist spinning of a classical knot by Zeeman in 1965. In the more recent developments, we follow the survey paper by G. Friedman, 2004, and we will generalize this to a knotted graph which is our recent results.
This is a joint work with Jozef H. Przytycki.
Date received: May 7, 2014
Copyright © 2014 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.