Topology Atlas | Conferences


Knots in Washington XXXIV; Categorification of Knots, Quantum Invariants and Quantum Computing
March 14-16, 2012
George Washington University
Washington, DC, USA

Organizers
Valentina Harizanov (GWU),Mark Kidwell (U.S. Naval Academy and GWU), Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Radmila Sazdanovic (U.Penn), Alexander Shumakovitch (GWU), Hao Wu (GWU)

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Abstracts

Meta-Groups, Meta-Bicrossed-Products, and the Alexander Polynomial
by
Dror Bar-Natan
University of Toronto

A straightforward proposal for a group-theoretic invariant of knots fails if one really means groups, but works once generalized to meta-groups (to be defined). We will construct one complicated but elementary meta-group as a meta-bicrossed-product (to be defined), and explain how the resulting invariant is a not-yet-understood generalization of the Alexander polynomial, while at the same time being a specialization of a somewhat-understood üniversal finite type invariant of w-knots" and of an elusive üniversal finite type invariant of v-knots".

Date received: March 1, 2012


Introduction to n-racks and their homology
by
Guy Roger Biyogmam
Southwestern Oklahoma State University

The category of n-racks has been recently introduced as a generalization of the category of racks to n-ary operations. In this talk, we present their motivation and discuss some properties. A cohomology theory on n-racks will be discussed as well.

Date received: February 7, 2012


Computing Distances in Graphs
by
Wesley Calvert
Southern Illinois University
Coauthors: Russell Miller and Jennifer Chubb Reimann

How can we compute the distance between vertices in a graph, given only data on adjacency? If there are infinitely many vertices, this problem may be unsolvable, but it can still be approximated in an interesting way.

It turns out that distances in graphs capture this sort of approximation exactly, in that any function that can be approximated can be approximated by a graph. I'll give examples that aren't obviously graph-like.

Date received: March 8, 2012


Smooth Cosmic Censorship
by
Vladimir Chernov
Dartmouth College
Coauthors: Stefan Nemirovski

It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard R4. Similarly, a smooth 4-manifold homeomorphic to the product of a closed oriented 3-manifold N and R and admitting a globally hyperbolic Lorentz metric is in fact diffeomorphic to N×R. Thus one may speak of a censorship imposed by the global hyperbolicty assumption on the possible smooth structures on (3+1)-dimensional spacetimes.

Date received: January 31, 2012


Generalized Hecke Algebras with Applications to Knot Theory.
by
Ben Cooper
University of Virginia

(Work in preparation). Hecke algebras have been closely tied to invariants of knots and links since the foundational work

of Jones. I will describe new, geometrically defined, families of these algebras and explain how they can be used to enrich

the story surrounding quantum invariants.

Date received: February 28, 2012


Torsion in 1-term Distributive Homology
by
Alissa S Crans
Loyola Marymount University
Coauthors: Jozef H. Przytycki, Krzysztof Putyra

In recent years, operations satisfying self-distributivity, (a * b) * c = (a * c) * (b * c), have secured an important role in knot theory. Sets equipped with such operations are known as "shelves".

Przytycki and Sikora proposed a conjecture about the homology of a certain class of shelves, which we proved. We will discuss the conjecture and its proof.

Date received: March 14, 2012


Vassiliev invariants from parity mappings
by
Heather A. Dye
McKendree University

A parity mapping assigns a weight to each chord in a Gauss diagram. The parity of the chords is used to construct invariants. In particular, we can construct a family of Vassiliev invariants.

Date received: March 8, 2012


Categorification of Generalized Jones-Wenzl Projectors
by
Matt Hogancamp
University of Virginia
Coauthors: Benjamin Cooper

The projections of tensor powers of the fundamental representation of quantum sl_2 onto the symmetric powers are called the Jones-Wenzl projectors. These have important applications to topology and have been categorified by previous authors. In this talk, we discuss a categorification of the entire decomposition of these tensor powers into irreducibles, and applications. This is joint work with Ben Cooper.

Date received: March 1, 2012


Twisted Alexander polynomials of 2-bridge knots
by
Jim Hoste
Pitzer College
Coauthors: Patrick D Shanahan

We develop a method to compute the twisted Alexander polynomial associated to dihedral representations of 2-bridge

knots. For several in nite classes of 2-bridge knots we use this method to verify a conjecture of Hirisawa and Murasugi

on the form of the resulting polynomial.

Date received: March 13, 2012


NilHecke algebra and its friends
by
Mikhail Khovanov
Columbia University

NilHecke algebra emerges from the geometry of flag variety and can be used to categorify parts of quantum sl(2) and its finite-dimensional representations. We'll review this theory and consider various modifications, including odd nilHecke algebras and their dg versions.

Date received: March 12, 2012


Krawtchouk matrices: interpretations and applications
by
Jerzy Kocik
Department of Mathematics, Southern Illinois University, Carbondale, IL62901

Krawtchouk matrices are related to Sylvester-Hadamard matrices used in quantum computing. We shall discuss their various interpretations and applications.

Date received: March 12, 2012


The Pure Virtual Braid Group is Quadratic
by
Peter Lee
University of Toronto

If an augmented algebra K over Q is filtered by powers of its augmentation ideal, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a universal finite type invariant.

Date received: February 25, 2012


Quantum Knots and Quantum Braids
by
Samuel J. Lomonaco
UMBC
Coauthors: Louis H. Kauffman

In this talk, we show how to reconstruct knot theory in such a way that it s intimately related to quantum physics. In particular, we give a blueprint for creating a quantum system that has the dynamic behavior of a closed knotted piece of rope moving in 3-space. Within this framework, knot invariants become physically measurable quantum observables, knot moves become unitary transformations, with knot dynamics determined by SchroedingerÂ’s equation. The same approach can also be applied to the theory of braids.

Toward the end of the talk, we briefly look at possible applications to superfluid vortices and to topological quantum computing in optical lattices.

Date received: March 6, 2012


Retractive groups
by
Keye Martin
Naval Research Laboratory

One goal of our current research is to define a new area called algebraic information theory. It began with the realization that many important classes of channels, both quantum and classical, possess the structure of a compact affine monoid. The idea is then to use this structure as the basis for new techniques in information theory.

In practice, many classes of channels arise as the convex closure of a certain underlying group, and among these, certain groups distinguish themselves in that they generate very restricted classes, which are remarkable in that they always contain the solution to optimization problems posed over the set of all channels. These groups, which we call retractive, can also be used to derive useful inequalities, devise methods for tomography of quantum channels and appear to always generate channels whose capacities can be expressed in closed form.

Date received: March 10, 2012


Embedding Groups into Distributive Subsets of the Monoid of Binary Operations
by
Gregory Mezera
George Washington University

Let X be a set and Bin(X) the set of all binary operations on X. We can make Bin(X) into a semigroup with the operation x*1*2y=(x*1y)*2y for all *1, *2 in Bin(X) and all x, y in X. We say that S ⊂ Bin(X) is a distributive set of operations if all pairs of elements *α, *β ∈ S are right distributive, that is, (a*αb)*βc = (a*βc)*α(b*βc) (we allow *α=*β). J.Przytycki raised the question of which groups can be realized as distributive sets. The initial guess that we may embed any group G into Bin(X) for some X was brought into question after Michal Jablonowski made an observation that if * ∈ S is idempotent (a*a=a), then * commutes with every element of S. In addition, Agata Jastrzebska computed all groups embedded in Bin(X) with |X| < =5, and found no nonabelian groups. However, the first noncommutative subgroup of Bin(X) (the group S3) was found computationally in October of 2011 by Yosef Berman. Here we show that any group can be embedded in Bin(X) for X=G (as a set). We do this by giving an explicit embedding we call the regular embedding, due to its relation with the regular representation of G. This embedding sends g to *g where
a*gb=ab-1gb.
Here, we check that the group {*g}g ∈ G is a distributive set: We have:
(a*g1b)*g2c = (ab-1g1b)*g2c = ab-1g1bc-1g2c and

(a*g2c)*g1(b*g2c)=(ac-1g2c)*g1(bc-1g2c) = ab-1g1bc-1g2c as needed.

We also discuss criteria for minimal embeddings of finite groups. That is, for a given group G, we show that the minimal |X| such that G embeds into a distributive subset of Bin(X) is related to the degree of the minimal faithful representation of G over F2, the field of two elements.

Paper reference: http://front.math.ucdavis.edu/1109.4850

Date received: March 9, 2012


Strategies in suppressing errors in quantum computing based in (quantum) group theory
by
Robert Owczarek
University of New Mexico and Enfitek, Inc.

I will briefly discuss the issue of control of errors in quantum computing and discuss in more detail approaches based in group theory and quantum group theory that serve this purpose.

Date received: March 2, 2012


Distributive homology: progress in the last two years
by
Jozef H. Przytycki
George Washington University

Let X be a set and Bin(X) the monoid of binary operations on X. We say that a subset S ⊂ Bin(X) is a distributive set if all pairs of elements *1, *2 ∈ S are right distributive, that is, (a*1b)*2c = (a*2c)*1(b*2c) (we allow *1=*2). We define a (one-term) distributive chain complex C(*) as follows: Cn=Z Xn+1 and the boundary operation ∂(*)n: Cn → Cn-1 is given by: ∂(*)n(x0, ..., xn) = (x1, ..., xn) + ∑i=1n(-1)i(x0*xi, ..., xi-1*xi, xi+1, ..., xn). The homology of this chain complex is called a one-term distributive homology of (X, *) (denoted by Hn(*)(X)). For a distributive set (*1, *2, ..., *k), the multi-term distributive homology Hn(a1, ..., ak)(X) is defined as the homology given by a chain complex (Cn, ∂(a1, ..., ak)) where Cn=ZXn+1 and ∂(a1, ..., ak) = ∑i=1k ai(*i).

The definition is less then 2 years old (although modeled on rack and quandle homology) but I am glad to report substantial progress due to work of Y.Berman, A.Crans, M.Jablonowski, G.Mezera, K.Putyra, and A.Sikora.

Paper reference: http://front.math.ucdavis.edu/1105.3700, http://front.math.ucdavis.edu/1109.4850, http://front.math.ucdavis.edu/1111.4772, http://at.yorku.ca/c/b/e/k/13.htm

Date received: March 12, 2012


Simplicial approach to distributive homology
by
Krzysztof Putyra
Columbia University
Coauthors: Jozef H. Przytycki, Alissa Crans

Most algebraic homology theories are reflected by topological constructions: group homology H(G) is given by homology of the Eilenberg-MacLane space K(G, 1), two-term rack homology HR(X) is given by the space BX defined by Fenn and Rourke, examined later also by Clauwens and Eisermann. In my talk I will construct a space EX for any shelf X, which gives one-term distributive homology of X. The construction is very similar to EG for groups. The idea was already mentioned by Sikora, but he never used it for actual computation. During my talk I will recover from EX a few known already results about distributive homology of certain racks and strengthen others. One suprising result is existence of torsion, which I will try to explain.

I will start my talk with recalling briefly definitions of distributive operations and distributive homology. Then I will talk about simplicial sets, focusing on its connection to topology of CW-complexes. In this setting I will show how EG is constructed for a group and explain, why a similar construction should hold for racks and shelves in general. In the last 15min I will recover and strengthen a few results using this simplicial set and speculate about conjectured exponential growth of distributive homology.

Date received: March 12, 2012


On the category of groupoids.
by
Piotr Stachura
Warsaw University of Life Sciences

Usually groupoid is defined as a (small) category with invertible morphisms. Such a definition suggests that morphism of groupoids is just a functor. The alternative definition of groupoid due to Zakrzewski in terms of relations will be presented. It turns out that it is equivalent to the usual definition but approach to morphisms is different: morphisms are relations that preserve structure. This will be described and some examples will be presented in purely algebraic and differential geometric situations.

Date received: March 12, 2012


(Over(?))-simplified notes about the Khovanov-Lipshitz-Sarkar spectra
by
Oleg Viro
Stony Brook University

(Over(?))-simplified notes about the Khovanov-Lipshitz-Sarkar spectra.

Date received: March 13, 2012


Homology of a Small Category with Functor Coefficients
by
Jing Wang
George Washington University
Coauthors: Jozef Przytycki

We first introduce the definition of homology of a small category with functor coefficients. Let C be a small category (i.e. Ob(C) forms a set), and F: C → R-Mod be a covariant functor from C to the category of modules over a commutative ring R. We call a sequence of objects and morphisms x0 → x1 → ...→ xn an n-chain (formally n-chain in the nerve of the category).We define the chain group Cn to be direct sum of F(x0) where the sum is taken over all n-chains x0 → x1 → ...→ xn. The boundary operation ∂ is defined to be the alternating sum of face maps d0, d1, ..., dn where for t ∈ F(x0), d0(t;x0 → x1 → ...→ xn) = (F(x0 → x1)(t);x1 → ...→ xn) and for i > 0, di(t;x0 → x1 → ...→ xn) = (t;x0 → x1 → ...→ xi-1 → xi+1 → ...→ xn). A standard checking shows that ∂2=0. The yielded homology is called homology of category C with Coefficients in functor F.

In the case of a category where objects have a well-defined ßize" we may often build a chain complex in a simplified way. This is the case when we deal with the category of an abstract simplicial complex. Let K = (V; P) be an abstract simplicial complex with vertices V (which we order) and simplexes P. We consider K as a small category with Ob(K) = P and morphisms are inclusions. Let F be a covariant functor from Kop to the category of modules over a commutative ring R where Kop denotes the opposite category of K. We define the chain group Cn to be the direct sum of F(S) where he sum is taken over all n-dimensional simplexes S. The boundary operation ∂ is given by the alternating sum of face maps d0, d1, ..., dn where if S = (v0, v1, ..., vn) is an ordered simplex of dimension n then on F(S), di = F(S ⊃ (S-vi)). Again we have ∂2=0. Thus we have a second definition of homology of an abstract simplicial complex category. It turns out that for an abstract simplicial complex category, these two definitions give the same homology groups. This is a known result for specialists. However, an elementary proof will be given here geared towards non-specialists.

Finally, we give another example which realizes Khovanov homology as homology of a simplex with coefficients in a specified Khovanov functor.

Date received: March 10, 2012


Copyright © 2012 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.