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Exotic TQFT: a new developement
by
Marta Asaeda
University of California at Riverside
We describe a new development in constructing exotic TQFT.
Date received: December 10, 2007
Khovanov Homology of Links Embedded in I-bundles
by
Jeffrey Boerner
University of Iowa
M. Asaeda, J. Przytycki and A. Sikora introduced a theory for links embedded in I-bundles. A new way to view their theory is introduced for I-bundles over orientable surfaces. The elements of the chain groups are surfaces rather than diagrams which gives the calculation of the theory a much more topological feel. This is done by borrowing ideas from D. Bar-Natans work involving Khovanov Homology for tangles and cobordisms.
Date received: October 20, 2007
The Gram determinant of the type-B Temperley-Lieb algebra
by
Qi Chen
Winston-Salem State University
Coauthors: Jozef Przytycki
The n-th type-B Temperley-Lieb algebra is the Kauffman bracket skein module of the annulus with 2n marks on one boundary. It admitts a symmetric bilinear form. The Gram determinat Dn of this bilinear form is conjectured, by Dabkowski and Przytycki, to be equivalent to the determinant of the type-B matrix of chromatic joints invested by Rodica Simon. Barad gave a formula for Dn. In this talk we will provide a proof for this formula. The proof uses the connection between the Kaffman bracket skein module and the representaion theory of sl2.
Date received: November 29, 2007
Effectively closed sets and orderings on groups
by
Jennifer Chubb
George Washington University
A countable group G is computable if there is an algorithm to determine membership in G as a set, and an algorithm for multiplication on the group. G is left-orderable (bi-orderable) if there is a linear ordering of the elements of the group that is left-invarient (both left- and right-invarient). I will describe how the orderings of a countable group may be viewed as infinite paths through a binary tree, and how the orderings of a computable group correspond to paths in a computable binary tree. Taking the usual topology induced on the paths, we see that these sets are closed subsets of Cantor space, and in the computable case, we can think of them as effectively closed. The effectively closed sets have been extensively studied in computability theory, and I will describe some of the computability theoretic consequences for the spaces of orderings on groups.
Date received: December 6, 2007
Unified SO(3)-quantum invariants for rational homology 3-spheres.
by
Buehler Irmgard
Winterthurerstrasse 190, 8057 Zuerich, Switzerland
Coauthors: A. Beliakova, T. Le
In 2001, K. Habiro constructed for an integer homology 3-sphere M a unified invariant I_M(q) which, if evaluated at any root of unity, gives the SU(2) Witten-Reshetikhin-Turaev invariant of M at that root. In this talk we extend Habiro's construction to rational homology 3-spheres. More precisely, given a rational homology 3-sphere M with |H_1(M;Z)|=b and an odd divisor c of b, we construct a unified invariant I_{M,c}(q), which dominates the SO(3) WRT invariants of M at all roots of unity whose order r has (r,b)=c.
Date received: November 7, 2007
Categorifications in algebra and topology
by
Mikhail Khovanov
Columbia University
Categorifications lift n-dimensional topological quantum field theories to (n+1)-dimensional ones (in most cases n=2 or 3). Algebraic categorifications turn semisimple representations into Grothendieck groups of nonsemisimple abelian or triangulated categories. We'll discuss examples and interrelations between these two types of categorifications.
Date received: December 1, 2007
The historical significance of Herbert Seifert's paper 'Ueber das Geschlecht von Knoten'
by
Mark E. Kidwell
U.S. Naval Academy
In one remarkable paper, Seifert gave a combinatorial algorithm for spanning an orientable, singularity-free surface in a knot in the 3-sphere, defined a matrix that describes the homology relations of this embedded surface, displayed a new and efficient method of computing the Alexander polynomial from this "Seifert matrix", and showed that half the degree of the Alexander polynomial gives a lower bound for the genus of the "Seifert surface". He also used his matrix to demonstrate exactly which polynomials can be Alexander polynomials of knots, a feat as yet unduplicated for any of the post-Jones polynomials. He provided an example of an apparently knotted curve that has Alexander polynomial 1 and, in a final flourish, proved its knottedness using hyperbolic geometry. If time permits, we will give a hopelessly incomplete review of further developments that stemmed from Seifert's brilliant paper.
Date received: November 29, 2007
Chromatic algebra and applications
by
Slava Krushkal
University of Virginia
Coauthors: Paul Fendley
In this talk I will introduce the notion of the chromatic algebra, whose Markov trace is the chromatic polynomial of an associated graph, and I will describe its applications in combinatorics and quantum topology. In particular, we give a new proof of Tutte's chromatic polynomial relations at the golden ratio, and we provide generalizations, conjectured by Tutte, for all Beraha numbers.
Date received: November 14, 2007
Knots, Quandles, and the Constraint Satisfaction Problem
by
Robert W. McGrail
Bard College
Coauthors: Peter Golbus, Mona Merling, Ken Ober, Japheth Wood
This is an overview of the current state of the Quandle Dichotomy Project within the Laboratory for Algebraic and Symbolic Computation at Bard College. In particular, the relations between quandles and constraint languages, and constraint satisfaction and nondeterministic polynomial time will be explored. Through these connections, a natural notion for the computational complexity of knots will be constructed.
Date received: October 23, 2007
Intrinsically n-linked spatial graphs
by
Danielle O'Donnol
UCLA
Coauthors: G.C. Drummond-Cole (Stony Brook)
A graph G, is intrinsically linked if every embedding of G into R3 contains a nontrivial link. The study of intrinsically knotted and linked graphs is a recent area of knot theory. I will give a summary of the history of intrinsically linked graphs. A natural generalization of intrinsic linking is intrinsic n-linking. A graph G is intrinsically n-linked if every embedding of G into R3 contains a non-split n-component link. I will discuss some of my results about intrinsic n-linking in complete and complete bipartite graphs.
Date received: November 7, 2007
Orders on computable torsion-free abelian groups
by
Sarah Pingrey
The George Washington University
Abstract: A countable group is computable if its domain is a computable set and its group theoretic operation is computable. We examine complexity of orders on a computable torsion-free abelian (hence orderable) group G, using Turing degrees as a complexity measure. There are continuum many Turing degrees and they form an upper semilattice under Turing reducibility. All computable sets have Turing degree zero. It is easy to see that if G is of rank 1, then G has exactly two orders and they are computable. Solomon showed that if G has a finite rank greater than 1, then G has an order in every Turing degree. On the other hand, if G is of infinite rank, then G does not necessarily have a computable order, as shown by Downey and Kurtz.
Date received: December 4, 2007
Torsion in H^2, v(G)-2_A_2(G) and its applications to Khovanov homology of adequate diagrams.
by
Jozef H. Przytycki
George Washington University
Coauthors: Radmila Sazdanovic (GWU)
It has been conjectured by Alexander Shumakovitch (announced at Knots in Poland conference in 2003) that any link which is not a connected or disjoint sum of Hopf links and trivial links has a torsion in Khovanov homology. Shumakovitch demonstrated the conjecture for alternating links and Marta Asaeda and myself generalized it for a large class of adequate links (including strongly adequate links). Here we prove the conjecture for those + adequate links D, whose + adequate diagram D has an associated s+ state graph Gs+(D) with a cycle of length at least 3. In our work we approximate Khovanov homology of D by chromatic (Helme-Guizon-Rong) cohomology of Gs+(D). In particular, we prove that for a connected simple graph G of v vertices and cyclomatic number p1 tor H2, v(G)-2(G) is equal to Z2p1 for G bipartite and Z2p1-1 otherwise.
Date received: December 6, 2007
Clock moves and a combinatorial homology
by
Yongwu Rong
George Washington University
Coauthors: Kerry Luse
This talk is motivated by an attempt to construct the combinatorial Floer homology via clock moves. For each link diagram, we construct graded homology groups using Kauffman's state sum and clock moves for the Alexander polynomial. While these groups are sometimes invariant under Reidemeister moves, they are, unfortunately, not always invariant under these moves. Nonetheless, we have a graded homology theory for link diagrams which yields the Alexander polynomial when taking graded Euler characteristic. This is joint work with Kerry Luse.
Date received: December 6, 2007
The Bar-Natan skein module of the solid torus and the homology of (n, n) Springer varieties
by
Heather Russell
University of Iowa
The Bar-Natan skein module of three-manifolds arose from Bar-Natans work with the Khovanov homology of tangles and cobordisms. We will show that the Bar-Natan skein module of the solid torus with boundary curve system 2n copies of the longitude is isomorphic to the total homology of the Springer variety of complete flags in C2n stabilized by a fixed nilpotent operator with two Jordan blocks of size n. In order to do this, we employ techniques found in Khovanov's work with crossingless matchings and the topological space S̃.
Date received: October 24, 2007
The Teichmüller distance between finite index subgroups of PSL2(Z)
by
Dragomir Saric
The City University of New York, Queens College
Coauthors: Vladimir Markovic
For a given ε > 0, we show that there exist two finite index subgroups of PSL2(Z) which are (1+ε)-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any ε > 0 there are two finite regular covers of the Modular once punctured torus T0 (or just the Modular torus) and a (1+ε)-quasiconformal between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S ) of the punctured solenoid S under the action of the corresponding Modular group has the closure in T(S ) strictly larger than the orbit and that the closure is necessarily uncountable. This is a joint work with V. Markovic.
Date received: November 23, 2007
Quantizations of Character Varieties of the torus
by
Adam S Sikora
SUNY Buffalo
We will discuss three different quantizations of character varieties of the torus. We will use these constructions to relate quantum invariants of knots to the topology of knot complements.
Date received: December 2, 2007
When Knots Don't Fiber
by
Dan Silver
University of South Alabama
Coauthors: Susan Williams (University of South Alabama)
In this joint work with Susan Williams we consider the conjecture: a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove it for a class of knots that includes all knots of genus 1. We also discuss two equivalent forms of the conjecture, one involving twisted Alexander polynomials, the other a weak form of subgroup separability.
Date received: December 4, 2007
Surgery description of colored knots
by
Steven Wallace
Louisiana State University
Coauthors: Richard A. Litherland
The pair (K,r) consisting of a knot K and a surjective map r from the knot group onto a dihedral group is said to be a p-colored knot. Moskovich conjectured that for any odd prime p there are exactly p equivalence classes of p-colored knots up to surgery along unknots in the kernel of the coloring. We show that there are at most 2p equivalence classes. This is a vast improvement upon the previous results by Moskovich for p=3, and 5, with no upper bound given in general. T. Cochran, A. Gerges, and K. Orr, in "Dehn surgery equivalence relations of 3-manifolds", define invariants of the surgery equivalence class of a closed 3-manifold M in the context of bordisms. By taking M to be 0-framed surgery of the 3-sphere along K we may define Moskovich's colored untying invariant in the same way as the Cochran-Gerges-Orr invariants. This bordism definition of the colored untying invariant will be then used to establish the upper bound.
Date received: September 27, 2007
Matrix factorizations and colored MOY graphs
by
Hao Wu
George Washington University
I will explain how to assign matrix factorizations to colored MOY graphs, which leads to a generalization of the Khovanov-Rozansky graph homology. I will also establish some basic properties of such matrix factorizations. This is a work in progress. Hopefully, it will lead to a categorification of the colored skein sl(n)-invariants.
Date received: December 9, 2007
Copyright © 2007 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.