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The topology of non-locality and contextuality
by
Samson Abramsky
Oxford University Computing Laboratory
Coauthors: Adam Brandenburger
Bell's theorem famously shows that no local theory can account for the predictions of quantum mechanics; while the Kochen-Specker theorem shows the same for non-contextual theories. Non-locality, and increasingly also contextuality, play an important role as computational resources in current work on quantum information. Much has been written on these matters, but there is surprisingly little unanimity even on basic definitions or the inter-relationships among the various concepts and results. We use the mathematical language of sheaves and monads to give a very general and mathematically robust description of the behaviour of systems in which one or more measurements can be selected, and one or more outcomes observed. In particular, we give a unified account of contextuality and non-locality in this setting.
- A central result is that an empirical model can be extended to all sets of measurements if and only if it can be realized by a factorizable hidden-variable model, where factorizability subsumes both non-contextuality and Bell locality. Thus the existence of incompatible measurements is the essential ingredient in non-local and contextual behavior in quantum mechanics.
- We give hidden-variable-free proofs of Bell style theorems.
- We identify a notion of strong contextuality, with surprising separations between non-local models: Hardy is not strongly contextual, GHZ is.
- We interpret Kochen-Specker as a generic (model-independent) strong contextuality result.
- We give general combinatorial and graph-theoretic conditions, independent of Hilbert space, for such results.
Paper reference: arXiv:1102.0264
Date received: April 23, 2011
The Geometry of Undecidability
by
Wesley Calvert
Southern Illinois University
Coauthors: Valentina Harizanov and Alexandra Shlapentokh
It has been known since the Ph.D. thesis work of Linda Richter in the 1980s that algebraic structures - groups, for instance - can encode degrees of undecidability in an intrinsic (isomorphism-invariant) way. From more recent work, we know that structures arising from algebraic geometry, including schemes, have a similar capacity. The present talk will describe these constructions.
Date received: April 28, 2011
Trivalent graphs and the Kauffman polynomial
by
Carmen Caprau
California State University, Fresno
We use trivalent graphs to construct a model for the 2-variable Kauffman polynomial for links. If time permitting, we will show how one can use this model to obtain an invariant of a certain type of trivalent graphs embedded in R3.
Date received: April 18, 2011
Theory of Quandles
by
J. Scott Carter
University of South Alabama
Coauthors: Masahico Saito
I will give the definition of a G-family of quandles that was introduced by Ishii, Iwakiri, Jang, and Oshiro. Given such a family for a fixed group, G, there is a quandle structure on the set X ×G where X is the underlying set of the G-family. There are natural ways of coloring knotted trivalent oriented spacial graphs and embedded foams in 4-space by G-families of quandles.
The relations induced by moves to these objects suggests a cohomology theory that incorporates classical group cohomology and quandle homology. The singularities that represent the graph and foam moves are dual to prismatic sets. Consequently, it is very easy to see that the boundary maps square to zero. I will sketch the definition of cocycle invariants associated to foams and to spacial graphs.
Date received: April 22, 2011
Causality in spacetimes and Legendrian linking
by
Vladimir Chernov
Dartmouth College
Coauthors: Stefan Nemirovski
Two points in a spacetime X are said to be causally related if one can get from one to the other travelling at less or equal than light speed. Low conecjture and the Legendrian Low conjecture formulated by Natário and Tod say that for nice spacetimes X two events x, y in X are causally related if and only if the link of spheres S_x, S_y whose points are light rays passing through x and y is non-trivial in the contact manifold N of all light rays in X.
We prove the Low and the Legendrian Low conjectures and show that similar statements are in fact true in almost all 4-dimensional globally hyperbolic spacetimes.
If time permits we discuss which of the smooth 4-manifolds admit a globally hyperbolic Lorentz metric, generalizing the results of Newman and Clarke.
Date received: March 28, 2011
From topology, via quantum entanglement, to arithmetic.
by
Bob Coecke
Oxford University
Coauthors: Aleks Kissinger and Alex Merry
(0) There is a canonical correspondence between certain symmetric states (i.e. vectors in C^n (x) C^n (x) C^n, or mor abstractly, morphisms of type I -> A (x) A (x) A in soem monoidal category) and arbitrary commutative Frobenius algebras CFA) on C^n (or A).
(1) In particular, for n=2 there are only two kinds of CFAs and they precisely correspond with the two kinds of non-trivial tripartite entanglement that exist in nature, so-called GHZ and W entanglement. The distinction between the CFAs is that the loop is either connected or disconnected.
(2) When passing from the Bloch sphere representation of a qubit to the stereographic projection, the CFAs are in fact nothing but multiplication and addition. In other words, up to `local transformations' all CFAs on C^2 are either addition or multiplication.
(3) Putting (1) and (2) together we obtain a topological interpretation of addition and multiplication, merely in terms of connectedness. We expect that this fact must have been observed in other contexts and are eager to learn about this.
Paper reference: In part based on arXiv:1002.2540 and arXiv:1103.2812
Date received: April 17, 2011
Representations of the Kauffman bracket skein algebra of the punctured torus
by
Razvan Gelca
Texas Tech University
Coauthors: Jea-Pil Cho
We examine a family of representations of the Kauffman bracket skein algebra of the punctured torus on skein modules of the solid torus with marked points on the boundary. We show how the Reshetikhin-Turaev representation of the mapping class group of the punctured torus can be computed using these representations.
Date received: April 14, 2011
Towards Categorification in Applied Mathematics
by
Robert Ghrist
Penn [Mathematics and Electrical/Systems Engineering
This talk will be a light survey of recent and current work in applications of algebraic topology to data, networks, and engineering systems. The emphasis will be on the (at present, inchoate) use of categorification as a theme for doing applied mathematics.
Date received: April 19, 2011
A relation between Khovanov homology and Kirby moves
by
Noboru Ito
Waseda University
This talk is a report on a relation between the Khovanov homology consisting of the enhanced Kauffman states and the Kirby moves.
Date received: April 7, 2011
Gauge Twists and Hennings TQFT's
by
Thomas Kerler
The Ohio State University
Coauthors: Qi Chen
The notion of a gauge twist of the co-algebra structure of a (strict) quasi-triangular Hopf algebra goes back to Drinfeld (1987). Such Hopf algebras also serve to construct TQFT's by extension of the Hennings calculus. This naturally raises the questions how TQFT's constructed from gauge twist equivalent Hopf algebra are related.
We show that these TQFT's are isomorphic and explicitly construct the natural isomorphism of TQFT-functors from the gauge twist tensor. A useful example arises from the observation that the double of the quantum-sl_2 Borel subalgbera is gauge twist equivalent to the tensor product of quantum-sl_2 and a copy of the Cartan subalgebra. We apply this in our forthcoming paper to prove integrality of associated quantum invariants.
Date received: April 14, 2011
Categorification of quantum groups
by
Mikhail Khovanov
Department of mathematics, Columbia University
The NilHecke algebra and its friends will be introduced and their importance for categorification of quantum groups explained.
Date received: April 11, 2011
"What you can tell about a link from the way it intersects a ball"; based on " An Obstruction to Embedding 4-Tangles in Links"
by
David A. Krebes
Calgary, Alberta CANADA
We consider the ways in which a 4-tangle T inside a unit cube can be extended outside the cube into a knot or link L. We present two links n(T) and d(T) such that the greatest common divisor of the determinants of these two links always divides the determinant of the link L. In order to prove this result we give a two-integer invariant of 4-tangles. Calculations are facilitated by viewing the determinant as the Kauffman bracket at a fourth root of -1, which sets the loop factor to zero. For rational tangles, our invariant coincides with the value of the associated continued fraction. In a modern development, we discuss the sign of the fraction for an alternating tangle.
Paper reference: math.GT/9902119
Date received: March 9, 2011
Invariants for knots in circle bundles
by
Joan Licata
Stanford University
Coauthors: Josh Sabloff
Many knot invariants are computed from a projection; for knots in Seifert fibered spaces, the natural projection is to a two-dimensional orbifold. We use such a diagram to construct rational Seifert surfaces and compute linking numbers. When the ambient manifold is a contact Seifert fibered space, this approach allows us to compute rational classical invariants and combinatorial Legendrian contact homology.
Date received: April 18, 2011
BiKei and unoriented link invariants
by
Sam Nelson
Claremont McKenna College
Coauthors: Sinan Aksoy
We define involutory biracks and biquandles (also known as BiKei) and use them to define counting invariants and enhanced counting invariants of unoriented knots and links. As an application, we will see a non-involutory biquandle whose counting invariant detects the non-invertibility of a virtual knot, answering a question of Xiao-Song Lin.
Paper reference: arXiv:1102.1473
Date received: March 21, 2011
Is quantum computing with superfluid helium possible: a potential role of Khovanov homology
by
Robert Owczarek
Enfitec, Inc.
Coauthors: L. H. Kauffman, J. H. Przytycki
Khovanov homology and related knot invariants open new ways for studying quantum statistical mechanics. One system of potential interest is superfluid helium. Namely, critical superfluid helium is a tangle of vortex loops, requiring description in terms of knot theory, which was never actually done, despite a number of attempts. My recent discussions with knot theorists showed some research directions that look promising. I am going to present a very preliminary discussion of the framework of the description of superfluid helium vortices in terms of Khovanov homology. I am also going to disuss shortly potential application of this system for quantum computing, in particular of the issue of which natural unitary operators are describing genuine quantum processes.
Date received: March 21, 2011
Weak and very weak simplicial modules: application to homology of distributive structures
by
Jozef H. Przytycki
George Washington University
In classical homological algebra a simplicial module (Mn, di, si) is a collection of
R-modules Mn, n ≥ 0, together with face maps di:Mn→ Mn-1 and degenerate maps
si: Mn→ Mn+1, 0 ≤ i ≤ n, which satisfy the following properties:
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If (X, *) is a right selfdistributive magma, called by Alissa Crans a shelf (i.e. (a*b)*c=(a*c)*(b*c)) then we define (motivated by Fenn, Rourke and Sanderson, and Carter, Kamada, and Saito), a presimplicial module (Mn, di), where Mn=RXn+1, d0(x0, x1, ..., xn) = (x1, ..., xn) and for i > 0 di(x0, x1, ..., xn)=(x0*xi, ..., xi-1*xi, xi+1, ..., xn). (We shift by 1 chain modules of FRS-CKS; that is Cn+1(X)=Mn(X)=RXn+1, to conform to the classical definition of a presimplicial module). The homology Hn(*)(X) of the chain complex (Mn, ∂n) are called one term distributive homology and their computation is a topic of a joint paper with Adam Sikora.
We can define degeneracy maps si: Mn→ Mn+1 by si(x0, ..., xn)=(x0, ..., xi-1, xi, xi, xi+1, ..., xn),
and notice that (Mn, di, si) satisfies axioms (1)-(3) of simplicial set. We can still consider
the submodules MDn ⊂ Mn defined by MDn=span(s0Mn-1, ..., sn-1Mn-1). We check
that (MDn, ∂n) is a subchain complex of (Mn, ∂n) iff
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In the case of spindles the homology splits: Hn(*)(X) = HnD(X) ⊕HQn(X) where HQn(X)=Hn(Mn/MnD). We ask whether the split holds generally for a weak simplicial module (recall that for a simplicial module, HDn(X)=0.). Assume now that only conditions (1)-(3) hold. In such a case we call (Mn, di, si) a very weak (frail) simplicial module. MnD is not necessary a subchain complex of (Mn, ∂n). To remedy this we consider level maps ti: Mn → Mn defined by ti=disi - di+1si and define boundary preserving filtration 1 ⊂ Ft0 ⊂ ... ⊂ Ftn-2 ⊂ Ftn-1=Mtn, and 1 ⊂ FtD0 ⊂ ... ⊂ FtDn-2 ⊂ FtDn-1=MtDn, by Fpt=FptMn = span(t0Mn, ..., tpMn), 0 ≤ i ≤ n, and FtDp = span(Ftp, s0Mn-1, ..., spMn-1) for p < n, FtDn = (FtDn-1, tnMn). We propose to call (MntD, ∂n) a generalized degenerate subchain complex in the theory of very weak simplicial modules and use it in calculations of homology of shelfs and racks. For example, we use it (or precisely F0t ⊂ FtD0 ⊂ Mn), to compute one term homology Hn*g(X) where a*gb=g(b) and g2=g. For a more general case of retraction of shelf r:X→ A, we compute one term homology in a paper with A.Sikora.
Paper reference: http://at.yorku.ca/c/b/b/r/24.htm
Date received: April 19, 2011
Mirror knots have dual odd Khovanov homology
by
Krzysztof Putyra
Columbia University
Coauthors: Wojciech Lubawski
Polish Academy of Sciences
An odd Khovanov homology is a modification of sl_2 link homology defined in 2008 by P. Ozsvath and Z. Szabo. Instead of a symmetric algebra they used an antisymmetric one and got a different link homology theory that categorifies the Jones polynomial. One year later I generalized both constructions to a theory with three parameters. Together with W. Lubawski we were able to find two gradings in the cube of resolutions, each splitting the cube into isomorphic pieces. This can be used to eliminate two parameters. Hence, up to isomorphism only two theories exist. As a consequence, mirror knots have dual chain complexes in both cases.
Date received: April 28, 2011
Comparing Quantum Invariants of 3-Manifolds
by
Matt Sequin
The Ohio State University
Coauthors: Thomas Kerler (Advisor)
We will describe two quantum 3-manifold invariants: The Kuperberg Invariant and the Hennings Invariant. We will discuss a direct proof that if H is an involutory Hopf algebra, the Kuperberg Invariant applied to H is equal to the Hennings Invariant applied to D(H), the Drinfeld double of H. We will then briefly discuss the case when H is not involutory.
Date received: April 7, 2011
Distributive products and their homology
by
Adam S Sikora
SUNY Buffalo
Coauthors: J. H. Przytycki
We discuss sets with distributive products (called shelves) and their homology, which underlies that of quandle and racks. Very little is known about the homology of shelves. We state number of open problems in that area.
Date received: April 29, 2011
Shift automorphisms of free groups and homeomorphisms of graphs
by
Ted Turner
George Washington University
Coauthors: Venu Addepali
Ben Atchison
A SHIFT AUTOMORPHISM of a free group is the analogue of a linear transformation whose RATIONAL CANONICAL FORM has a single block. We address the question of which automorphisms of finite order (or finite outer order) are shift automorphisms. In the process, we develop some useful invariants for HOMEOMORPHISMS OF FINITE GRAPHS.
Date received: April 22, 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.