Topology Atlas | Conferences

Knots in Washington XXVI Interconnections between Khovanov, Khovanov-Rozansky and Ozvath-Szabo homology, categorification of skein modules
April 18-20, 2008
George Washington University
Washington, DC, USA

Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Alexander Shumakovitch (GWU), Hao Wu (GWU)

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Cobordism of fibered knots, applications to low dimensional algebraic knots.
V. Blanloeil
Louis PASTEUR University of Strasbourg

In this talk we will explain the classical theory of knot cobordism (low and high dimensions). Then we will give some applications to cobordism of knots associated with isolated singularities of complex hypersurfaces; in particular we will focus on Brieskorn singularities.

Date received: March 27, 2008

Khovanov homology in thickened surfaces
Jeffrey Boerner
University of Iowa

I will introduce an infinite family of link homologies in thickened orientable surfaces. One member of the family is closely related to the homology theory developed by M. Asaeda, J. Przytycki and A. Sikora. In addition I will introduce a homology theory whose graded euler characteristic is exactly the Kauffman Surface Bracket.

Paper reference: arXiv:0801.3227

Date received: March 27, 2008

Polyak-Viro formulas for coefficients of the Conway polynomial.
Sergei Chmutov
The Ohio State University, Mansfield

I will report a work of my students Michael (Cap) Khoury and Alfred Rossi who found Polyak-Viro arrow diagram formulas for the coefficients of the Conway polynomial. The formulas are related to a state model of F.Jaeger and answer one of his questions. This model gives a description of the Conway polynomial as a state sum over some subsets of crossings of a knot diagrams.

Date received: March 19, 2008

Ozsváth-Szabó and Rasmussen invariants of cable knots
Cornelia Van Cott
Indiana University

We study the behavior of the Ozsváth-Szabó and Rasmussen concordance invariants on cable knots. In the case of the Ozsváth-Szabó invariant, we extend results of Hedden, finding bounds on the value of the invariant and computing the invariant explicitly for all cables of a particular class of knots.

Date received: March 27, 2008

Turaev surfaces and the determinant of a knot
Oliver Dasbach
Louisiana State University

Turaev constructed to each knot diagram an oriented surface on which the knot projects alternatingly. The Jones polynomial can now be computed from the Bollobas-Riordan-Tutte polynomial of one of the two checkerboard graphs of the knot projection on this surface. In particular the determinant of the knot has an appealing interpretation in this setting.

Date received: April 8, 2008

Nonuniform Thickness and Weighted Distance
Oguz C Durumeric
University of Iowa

Non-uniform tubular neighborhoods of submanifolds of the Euclidean space R^n are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtained for non-uniform thickness. All singularities within almost injectivity radius in dimension 1 are classified by the Horizontal Collapsing Property. Examples are provided to show the distinction between the different types of injectivity radii, as well as showing that the standard differentiable injectivity radius fails to be upper semicontinuous on a singular set of weight functions.

Paper reference: arXiv:0705.2407

Date received: February 12, 2008

Dynamical zeta functions, Floer homology and categorification
Alexander Fel'shtyn
University of Szczecin and Boise State University

We describe the connection between symplectic Floer homology of symplectomorphisms of surface and Nielsen-Thurston theory. Symplectic zeta functions are discussed. A project of a categorification of dynamical zeta functions is proposed. The first indication of a possibility of such categorification is a formula of Milnor's that relates Weil zeta function of a monodromy map to the Alexander polynomial and Ozsvath-Szabo categorification of the Alexander polynomial.

Date received: April 2, 2008

Parametrized Morse Theory on Foams and Network TQFT
Charles Frohman
The University of Iowa
Coauthors: Dennis Roseman

We analyze generic codimension zero and codimension one singularities of smooth functions on Foams.

A foam is a singular suface whose singularities are modeled on a cylinder over the letter Y. The facets of a foam are oriented and they induce a consistent orientations on the seams and the orientation on the boundary makes the boundary into an oriented trivalent graph whose vertices are sources or sinks. Finally, there is a choice of cyclic ordering of the facets coming up to any seam, which leads to a cyclic ordering of the edges coming up to a boundary web. The facets of a foam are colored.

A network TQFT assigns a vector space to any colored web, and a linear map to each foam.

We use the Morse theory of smooth functions on foams to derive basic data, and the consistency conditions on that data needed for a network TQFT to exist.

This research is inspired by the paper "Network TQFT" by Serge Natanzon. Our Foams are both more restrictive and less restrictive than his. We do not have vertices, yet our facets can have genus and his cannot.

Date received: March 25, 2008

Weak congruence and the quantum ideal of a 3-manifold
Patrick Gilmer
Louisiana State University

Fix an odd prime p. Let Ap denote a primitive 2pth root of unity and h = 1+Ap. Let Op denote Z[Ap]. The quantum ideal Jp(M) is the Op-ideal generated by the Ip-invariant of all links in a closed 3-manifold M. Then there is a non-negative integer ap(M) such that Jp(M) = (h)ap(M). ap is an invariant of weak p-congruence. Let c(M), cp(M), g(M) denote respectively the co-rank of π1(M), the p-cut number, and the Heegaard genus of M. Then c(m) ≤ cp(M) ≤ 2 ap(M)/(p-3) ≤ g(M).

Date received: April 1, 2008

On Floer homology and knots admitting lens space surgeries
Matthew Hedden
Coauthors: K. Baker, J.E. Grigsby

J. Berge discovered a simple condition on a knot, K, in the three-sphere which ensures that Dehn surgery on K yields a lens space. It is an open conjecture, known as the Berge conjecture, that any knot on which one can perform surgery and obtain a lens space satisfies his condition. I will discuss a strategy, developed jointly with Ken Baker and Eli Grigsby by which the knot Floer homology invariants of Ozsvath, Szabo, and Rasmussen could be used to prove this conjecture.

Paper reference: arXiv:0710.037, arXiv:0710.0359

Date received: April 8, 2008

Smooth spaces: convenient categories for differential geometry
Alexander Hoffnung
University of California, Riverside
Coauthors: John Baez

In 1977 K.T. Chen introduced a notion of smooth spaces as a generalization of the category of smooth manifolds. In 1979 Souriau introduced another notion, 'diffeological spaces', serving the same purposes. Both of these categories have all limits and colimits, and are cartesian closed. In fact, following ideas of Dubuc, we give a unified proof that the categories of Chen spaces, diffeological spaces, and simplicial complexes are 'quasitopoi': locally cartesian closed categories with finite (and in these cases all) colimits and a weak subobject classifier.

Date received: March 18, 2008

Frobenius algebras and skein modules of surfaces in 3-manifolds
Uwe Kaiser
Boise State University

For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skein module of the 3-ball is isomorphic to the ground ring of the Frobenius algebra. We discuss a presentation theorem for the skein module with generators incompressible surfaces colored by elements of a generating set of the Frobenius algebra, and with relations determined by tubing geometry in the 3-manifold and relations of the algebra. We discuss some ideas how the structure of the module depends on the 3-manifold (surgery) and how it changes under deformations of the Frobenius algebra (Frobenius manifolds).

Paper reference: arXiv:0802.4068

Date received: April 3, 2008

How to categorify a quantum group
Mikhail Khovanov
Columbia University and IAS
Coauthors: Aaron Lauda

I'll explain a recent joint work with Aaron Lauda on categorification of one-half of the quantum universal enveloping algebra associated to a simply-laced Dynkin diagram. If time allows, Lauda's work on categorification of the entire quantum sl(2) will be reviewed as well.

Date received: March 27, 2008

Knot Floer width and Turaev genus
Adam Lowrance
Louisiana State University

The width of a bigraded knot homology theory is the maximum distance plus one between slope one diagonals with respect to the bigrading that support the group. The Turaev surface of a knot diagram is obtained by associating a canonical ribbon graph to that diagram. Turaev genus is the minimum genus Turaev surface for all diagrams of the knot. We show that Turaev genus gives a natural bound for width of knot Floer homology.

Paper reference: arXiv:0709.0720

Date received: April 7, 2008

Khovanov-Rozansky link homology using foams
Marco Mackaay
Universidade do Algarve
Coauthors: Marko Stosic and Pedro Vaz

In my talk I will explain how to use foams to obtain a alternative construction of Khovanov-Rozansky link homology. Our goal was to find a completely combinatorial approach. Unfortunately we have not achieved that goal yet, but we have made some progress which I will explain.

Paper reference: arXiv:0708.2228

Date received: April 7, 2008

The Khovanov and Floer homologies of quasi-alternating knots
Ciprian Manolescu
Columbia University
Coauthors: Peter Ozsvath

We show that the Khovanov and knot Floer homologies are thin for a large class of knots, called quasi-alternating.

Paper reference: arXiv:0708.3249

Date received: March 7, 2008

A Diagram-Free Link Homology
Adam McDougall
University of Iowa

Link homologies are typically dependent on diagrams. That is, one usually gets a homology from the diagram corresponding to the projection of a particular link, then tries to prove that the homology is invariant under the 3 Reidemeister moves. Having a homology that is created from a true equivalent class of links under isotopy in 3-space has the advantage of bypassing the need for the 3 Reidemeister moves. In my work I set up a chain complex from a given link and study its homology, comparing it to other well-known link homologies.

Date received: March 27, 2008

Trivalent Graphs and the HOMFLYPT Polynomial
Annie Meyers
Graduate Student, University of Iowa
Coauthors: Charlie Frohman (advisor, University of Iowa)

I am currently working with Charlie Frohman, focusing on various state-sum formulas for the HOMFLYPT polynomial. The two I have been working with the most are a setup using 4-valent graphs by Jaeger and Kauffmann, and a setup using trivalent graphs by Murakami, Ohtsuki, Yamada, and expanded further by Rasmussen. My hope is to take my understanding of the HOMFLYPT and these state-sum formulas and apply it to working with foams and link homologies. I am currently in preparation for my comprehensive exam, focusing on the Jaeger/Kauffmann and MOY/Rasmussen ideas, and plan to continue research in this vein after finishing the exam.

Date received: March 27, 2008

Quantum Hyperbolic Invariants
Heather Molle
University of Iowa

My research concerns the volume conjecture. I am studying the invariants of Baseilhac and Bennedetti as well as their growth. In particular, I am computing these invariants for the figure eight knot and its double branched cover.

Date received: April 4, 2008

Enhancements of Counting Invariants
Sam Nelson
Pomona College

The number of colorings of a link diagram by finite quandle is an easily computable link invariant. We will look at a number of ways of strengthening these counting invariants for certain types of coloring quandles by exploiting the quandle's extra structure.

Paper reference: arXiv:0801.2979

Date received: March 2, 2008

Knot homologies and transverse knots
Lenny Ng
Duke University

I'll survey recent progress in the application of knot homologies (especially knot Floer homology) to the problem of distinguishing transverse knots in standard contact three-space.

Date received: March 30, 2008

Dehn surgeries that reduce the Thurston norm of a fibred manifold
Yi Ni
AIM/Columbia University

We study the Dehn surgeries that reduce the Thurston norm of the fibre class of a fibred manifold. It turns out that such surgeries are exactly the ones that are expected.

Date received: March 26, 2008

Minor Minimal Intrinsically 3-linked Graphs
Danielle O'Donnol

A graph G, is intrinsically linked if every embedding of G into R3 contains a non-split 2-component link. The study of intrinsically knotted and linked graphs is a recent area of knot theory. A natural generalization of intrinsic linking is intrinsic n-linking. A graph G, is intrinsically n-linked if every embedding of G in R3 contains a non-split n-component link. I will discuss some of my results about minor minimal intrinsically 3-linked graphs.

Date received: March 28, 2008

On generalization of odd Khovanov homologies
Krzysztof Putyra
Jagiellonian University, Krakow

In paper 'Odd Khovanov homology' P. Ozsvath, J. Rasmussen and Z. Szabo described a link homology related to Khovanov's theory. It is given by a projective functor from the category of (1+1)-cobordisms to the category of graded Z-modules. Giving cobordisms an additional structure we can give a functorial description of this construction. This leads into more general theory which both Khovanov's (with c=0) and odd homologies are special cases of. Moreover, one can prove the independence on the Reidemeister moves at the topological level, like in the Bar-Natan's paper 'Khovanov's homology for tangles and cobordisms'. In the talk I will describe this general construction, show the connection to known theories and, if time permits, give an idea of the proof of independence of the construction.

Date received: March 13, 2008

On higher order relations of nested arcs
Masahico Saito
University of South Florida

Nested arcs in the upper-half plane, or crossingless matchings, play important roles in Khovanov homology and (1+1)-TQFT. They also correspond to parenthesis structures, thus associativity and associahedron (the Stasheff polytope). In this talk, an analog of the process of constructing the associahedron from associativity is explored for the saddle points (surgery) of nested arcs. Higher order relations thus obtained have similarity to cohomology of Frobenius algebras.

Date received: April 9, 2008

Unknotting numbers of diagrams of a given nontrivial knot are unbounded
Kouki Taniyama
Waseda University and George Washington University

We show that for any nontrivial knot K and any natural number N there is a diagram D of K such that the unknotting number of D is greater than N.

Date received: April 5, 2008

Looking for non-zero maps between isomorphism classes of tangles
Robert Todd
University of Nebraska Omaha

A cobordism between two links L1 and L2 induces a map between the Kh(L1) and Kh(L2). Dror Bar-Natan gave a way to talk about cobordisms between tangles and the maps that they induce on the complexes associated to these tangles. Given two knots it is clear what are all possible non-zero maps from the Khovanov homology of one to the Khovanov homology other. Here we given a homology theory on surfaces which allows one to determine all possible chain maps between Bar-Natan's tangle complexes that induce non-zero maps on the isomorphism classes of the tangles. As an example we derive the maps associated to the Reidemeister moves.

Date received: March 27, 2008

Trip graphs of classical knots
Lorenzo Traldi
Lafayette College
Coauthors: Louis Zulli (Lafayette College)

A plane diagram of a classical knot yields a chord diagram, which in turn yields an intersection graph. If we modify this intersection graph by adjoining loops at the vertices that correspond to negative crossings, we obtain the "trip graph". Its adjacency matrix is the trip matrix, which yields the Jones polynomial [Zulli, Topology 34 (1995)]. This derivation of the Jones polynomial is quite different from the better-known derivation from the checkerboard graph, related to the Tutte polynomial [Thistlethwaite, Topology 26 (1987)]. Crossings give rise to edges in the checkerboard graph and vertices in the trip graph, for one thing. Also, the trip graph construction does not extend to links of more than one component, so it does not give rise to the usual recursive description of the Jones polynomial.

We discuss the invariants of arbitrary graphs that correspond to the Kauffman bracket and the Jones polynomial. They have several interesting properties. For instance: the graph bracket distinguishes nonisomorphic looped graphs with no more than 6 vertices, and also simple graphs with 7 vertices; the graph bracket may be calculated through an unfamiliar recursion that uses the pivot and local complementation operations associated with the interlace polynomial [Arratia, Bollobás and Sorkin, J. Combin. Theory Ser. B 92 (2004)]; and the graph Jones polynomial is invariant under graph-theoretic versions of the Reidemeister moves.

Date received: February 21, 2008

Twisted acyclicity of circle and link signatures
Oleg Viro
Stony Brook University & PDMI

A part of the classical link theory related to various signature invariants of link will be presented on the basis of twisted homology. This approach allows us to generalize most of results to higher dimensional codimension 2 links with components transversally intersecting each other.

Date received: April 12, 2008

A4-colored knots and their surgery equivalence
Steven D. Wallace
Louisiana State University

The pair (K, r) consisting of a knot in the three-sphere and a representation of the knot group onto the alternating group on four letters is said to be an A4-colored knot. We establish lower and upper bounds for the number of equivalence classes of A4-colored knots up to surgery along unknots representing elements in the kernel of r. Such surgeries preserve A4-colorability. We do this by defining a complete invariant for A4-colored surgery equivalence. Indeed, this is an analog to the classical result that every knot has a "surgery description" or equivalently that every knot is surgery equivalent to the unknot if we place fewer restrictions on the allowed surgery curves.

Date received: March 5, 2008

On Combinatorial Floer homology
J.J. Wang
California Institute of Technology

I will talk about the combinatorial description of the hat version Heegaard Floer homology.

Date received: April 1, 2008

Khovanov homology and lens space surgeries
Liam Watson
Université du Québec à Montréal

We study a notion of stability in Khovanov homology and derive obstructions to certain exceptional surgeries. This talk will focus on a particular example: Lens space surgeries on the 3-sphere.

Date received: April 7, 2008

Springer fibers and disoriented knot homology
Ben Webster
Institute for Advanced Study/MIT
Coauthors: Catharina Stroppel

We describe a geometric construction, using certain Springer fibers, of a modification of Khovanov's arc algebra. This modified arc algebra can be thought of as a "disoriented" analogue of the arc algebra, in the sense of the Morrison-Walker disoriented TQFT construction of knot homology. While still preliminary, this result suggests a connection between the Seidel-Smith symplectic knot homology and disoriented Khovanov homology.

Date received: March 28, 2008

A new categorification of the colored Jones polynomial
Stephan Wehrli
Columbia University

We describe an action of the n-th symmetric group on the disoriented Khovanov homology of the n-cable of a framed oriented knot in R^3. Using `spinorial confusions´, we then show that this action factors through an action of the n-th Temperley-Lieb algebra at q=1. Our results lead to a new categorification of the non-reduced colored Jones polynomial, and we prove that this categorification is essentially equivalent to Khovanov's categorification of the non-reduced colored Jones polynomial.

Date received: March 17, 2008

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