Topology Seminar

Seminars During Academic Year 2010-11

Fall 2010

Tuesday September 7, 2010 in Lindley Hall 019

Eiji Ogasa, Local moves of n-dimensional knots

We consider how $n$-knots are changed and how invariants of $n$-knots are changed if $n$-knots are changed by local moves of $n$-knots. In the $n=1$ case, some results are well known: pass-moves of 1-knots and the Arf invariants, crossing change of 1-knots and the Conway-Alexander polynomial (resp. the Jones polynomial, the Vassiliev invariants). We consider what kind of local moves of $n$-knots ($n\geqq2$) should be considered in order to generalize the $n=1$ case. This talk is mainly based on the speaker's published papers: [1]Local move identities for the Alexander polynomials of high dimensional knots and inertia groups. Journal of knot theory and its ramificatioms vol18, no.4, 2009 531-545, math.GT/0512168, UTMS 97-63. [2]Ribbon-moves of 2-links preserve the $\mu$-invariant of 2-links. Journal of knot theory and its ramificatioms, vol13, no5, 2004, 669-687, math.GT/0004008, UTMS 97-35. [3]Supersymmetry, homology with twisted coefficients and n-dimensional knots. International Journal of Modern Physics A Vol. 21, Nos. 19 & 20 (10 August 2006), pp.4185-4196, hep-th/0311136.

Wednesday September 8, 2010 in Rawles Hall 104

Allan Edmonds, Group actions on aspherical manifolds

This will be a survey of some topics related to finite group actions on aspherical manifolds, somewhat from the point of view of low dimensional topology.

Wednesday September 15, 2010 in Rawles Hall 104

Margaret Doig, Heegaard-Floer homology and knot surgery

Thursday September 23, 2010 in Rawles Hall 104

Heather Dye, Categorifications of the Arrow Polynomial

I will present several categorifications of the arrow polynomial, an invariant of virtual links. An overview of virtual links, the arrow polynomial, and Khovanov homology will be included.

Wednesday October 6, 2010 in Rawles Hall 104

Chuck Livingston, Bing Doubling and the boundary genus of links

Wednesday October 13, 2010 in Rawles Hall 104

Margaret Doig, Heegaard-Floer homology and knot surgery (continued)

Monday October 18, 2010 in Rawles Hall 316

Jonathan Yazinski, Involutions on exotic 4-manifolds

Wednesday October 20, 2010 in Rawles Hall 104

Nathaniel Rounds, Local Poincare Duality

Thursday October 21, 2010 in Rawles Hall 104

Qayum Khan, On isovariant rigidity of CAT(0) manifolds

We discuss Quinn's equivariant generalization of the Borel Conjecture. This concerns cocompact proper actions of a discrete group $\Gamma$ on a Hadamard manifold $X$. We give a complete solution when the action of $\Gamma$ is pseudofree and when $X$ more generally is a CAT(0) manifold. Here, emph{pseudofree} means that the singular set is discrete. A rich class of examples is obtained from crystallographic groups $Gamma$ made out of isometric spherical space form groups $G$. If $\Gamma$ has no elements of order two, then we obtain equivariant topological rigidity of the pair $(X,\Gamma)$. Hence, if $\Gamma$ is torsion-free, then we generalize a recent theorem of A.~Bartels and W.~L"uck, which validates the classical Borel conjecture for CAT(0) fundamental groups. Otherwise, if $\Gamma$ has elements of order two, we show how to parameterize all possible counterexamples, in terms of Cappell's UNil summands of the $L$-theory of infinite dihedral groups. These projects are joint work with Frank Connolly and Jim Davis.

Wednesday November 3, 2010 in Rawles Hall 104

Ben McCarty, An infinite family of Legendrian torus knots distinguished by cube number

For a knot K, the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. There is also Legendrian version of this invariant called the Legendrian cube number. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number

Tuesday November 9, 2010 in Rawles Hall 104

Joan Licata, Legendrian contact homology for Seifert fibered spaces

In this talk, I'll focus on Seifert fibered spaces whose fiber structure is realized by the Reeb orbits of an appropriate contact form. A Legendrian knot in such a manifold is described by a specially labeled Lagrangian diagram, and from this data one can compute both the "classical" invariants for Legendrian knots in rational homology three-spheres and also a new invariant which takes the form of a differential graded algebra. This work is joint with J. Sabloff.

Wednesday December 1, 2010 in Rawles Hall 104

Sam Lewallen, Twisted signatures and some topology behind the Jones polynomial

We'll discuss a new relationship between Khovanov homology and SU(2) representations of the knot group. In simple cases, this leads to a definition of the Jones polynomial directly in terms of the SU(2) representation variety. In particular, we show that for 2-bridge knots, the Jones polynomial is determined by certain invariants of flat connections on the knot complement, which are themselves equal to some classical, metacyclic invariants of the knot group.

Spring 2011

Wednesday February 9, 2011 in Rawles Hall 104

Jae Choon Cha, Applications of covering link calculus

TBA

Wednesday February 23, 2011 in Rawles Hall 104

Ryan Budney, Symmetry of knots

Abstract: In the 80's Allen Hatcher proved a rather remarkable conjecture of Steve Smale, that says that in principle there is a "uniform" way to straighten all unknots. _Specifically, the path-component of the trivial knot in the space of smooth embeddings of the circle in the 3-sphere has as a deformation-retract the subspace of great circles. _This talk will be about a certain logical extension of these ideas into families of canonical positions for all knot types, developing connections between operads and geometrization of 3-manifolds.

Wednesday March 2, 2011 in Rawles Hall 104

Ayelet Lindenstrauss, The Algebraic K-Theory of the Ring of Formal Power Series

(joint with Randy McCarthy) We study the algebraic K-theory of parametrized endomorphisms of a unital ring R with coefficients in a simplicial R-bimodule M, and compare it with the algebraic K-theory of the ring of formal power series in M over R. Waldhausen defined an equivalence from the suspension of the reduced Nil K-theory of R with coefficients in M to the reduced algebraic K-theory of the tensor algebra T_R(M). Extending Waldhausen's map from nilpotent endomorphisms to all endomorphisms, our map has to land in the ring of formal power series rather than in the tensor algebra, and is no longer in general an equivalence (it is an equivalence when the bimodule M is connected). Nevertheless, the map shows a close connection between its source and its target: it induces an equivalence on the Goodwillie Taylor towers of the two (as functors of M, with R fixed), and allows us to give a formula for the suspension of the invariant W(R;M) (which can be thought of as Witt vectors with coefficients in M, and is what the Goodwillie Taylor tower of the source functor converges to) as the inverse limit, as n goes to infinity, of the reduced algebraic K-theory of T_R(M)/ (M^n).

Wednesday March 9, 2011 in Rawles Hall 104

Adam Levine, A Combinatorial Spanning Tree Model for Knot Floer Homology

We provide an explicit description of a complex, based on spanning trees of the black graph of a diagram of a knot K in S^3, that computes the knot Floer homology of K. The strategy is to iterate Manolescu's unoriented skein exact sequence for knot Floer homology, using twisted coefficients in a power series ring, to form a cube of resolutions in which the only nonzero groups correspond to the connected resolutions. This construction has intriguing similarities with Ozsvath and Szabo's spectral sequence from the reduced Khovanov homology of K to the Heegaard Floer homology of the double branched cover of K, and it suggests an approach toward understanding the behavior of knot Floer homology under mutation. This is joint work with John Baldwin.

Thursday March 24, 2011 in Rawles Hall 104

Philip Hackney, Operations in the spectral sequence of a cosimplicial space

Wednesday March 30, 2011 in Rawles Hall 104

Palanivel Manoharan, Lefschetz fixed point theory on fibrations

Wednesday April 13, 2011 in Rawles Hall 104

Josh Greene, Lattices, graphs, and Conway mutation

I will discuss the proof and consequences of the following result: if D and D' are reduced, alternating diagrams for a pair of links with branched double covers Y and Y', then D and D' are mutants iff Y and Y' are homeomorphic iff Y and Y' have isomorphic Heegaard Floer homology. _The main input is a combinatorial result which characterizes the 2-isomorphism type of a graph in terms of the d-invariant of its lattice of integral flows.

Thursday April 14, 2011 in Rawles Hall 104

Mark Powell, Links not concordant to the Hopf Link

Wednesday April 20, 2011 in Rawles Hall 104

Pat Gilmer, Irreducibility properties of modular representations of mapping _class groups arising in TQFT

We find _decomposition series of length at most two for the modular representations of the mapping class groups of a one-holed surfaces induced by integral TQFT. We given the dimensions of these irreps.

Thursday April 21, 2011 in Rawles Hall 104

Tetsuya Abe, The Rasmussen invariant of a homogeneous knot

homogeneous knot is a generalization of alternating knots and positive knots. In this talk, we determine the Rasmussen invariant of a homogeneous knot. As a corollary, we obtain some characterizations of a positive knot. In particular, we recover Baader_s theorem which states that a knot is positive if and only if it is homogeneous and strongly quasipositive.

Wednesday April 27, 2011 in Rawles Hall 104

Yi-Jen Lee

Thursday April 28, 2011 in Rawles Hall 104

Tetsuya Abe, The Rasmussen invaiant of homogeneous links

Thursday May 5, 2011 in Rawles Hall 104

Matt Hedden, Contact Structures and Heegaard-Floer Homology