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Series: Geometry Topology Seminar

This is joint work with Mike Sullivan. We consider a Legendrian surface L in R5 or more generally in the 1-jet space of a surface. Such a Legendrian can be conveniently presented via its front projection which is a surface in R3 that is immersed except for certain standard singularities. We associate a differential graded algebra (DGA) to L by starting with a cellular decomposition of the base projection to R2 of L that contains the projection of the singular set of L in its 1-skeleton. A collection of generators is associated to each cell, and the differential is determined in a formulaic manner by the nature of the singular set above the boundary of a cell. Our cellular DGA is equivalent to the Legendrian contact homology DGA of L whose construction was carried out in this setting by Etnyre-Ekholm-Sullivan with the differential defined by counting holomorphic disks in C2 with boundary on the Lagrangian projection of L. Equivalence of our DGA with LCH is established using work of Ekholm on gradient flow trees. Time permitting, we will discuss constructions of augmentations of the cellular DGA from two parameter families of functions.

Series: Geometry Topology Seminar

A Jones surface for a knot in the three-sphere is an essential surface whose boundary slopes, Euler characteristic, and number of sheets correspond to quantities defined from the asymptotics of the degrees of colored Jones polynomial. The Strong Slope Conjecture by Garoufalidis and Kalfagianni-Tran predicts that there are Jones surfaces for every knot.
A link diagram D is said to be a Murasugi sum of two links D' and D'' if a state graph of D has a cut vertex, which separates the graph into two state graphs of D' and D'', respectively. We may obtain a state surface in the complement of the link K represented by D by gluing the state surface for D and the state surface for D' along the disk filling the circle represented by the cut vertex in the state graph. The resulting surface is called the Murasugi sum of the two state surfaces.
We consider near-adequate links which are certain Murasugi sums of near-alternating link diagrams with an adequate link diagram along their all-A state graphs with an additional graphical constraint. For a near-adequate knot, the Murasugi sum of the corresponding state surface is a Jones surface by the work of Ozawa. We discuss how this proves the Strong Slope Conjecture for this class of knots.

Series: Geometry Topology Seminar

A 2-knot is defined to be an embedding of S^2 in S^4. Unlike the theory of concordance for knots in S^3, the theory of concordance of 2-knots is trivial. This talk will be framed around the related concept of 0-concordance of 2-knots. It has been conjectured that this is also a trivial theory, that every 2-knot is 0-concordant to every other 2-knot. We will show that this conjecture is false, and in fact there are infinitely many 0-concordance classes. We'll in particular point out how the concept of 0-concordance is related to understanding smooth structures on S^4. The proof will involve invariants coming from Heegaard-Floer homology, and we will furthermore see how these invariants can be used shed light on other properties of 2-knots such as amphichirality and invertibility.

Series: Geometry Topology Seminar

John Etnyre: "Embeddings of contact manifolds"
Abstract: I will discuss recent results concerning embeddings and
isotopies of one contact manifold into another. Such embeddings should
be thought of as generalizations of transverse knots in 3-dimensional
contact manifolds (where they have been instrumental in the development
of our understanding of contact geometry). I will mainly focus on
embeddings of contact 3-manifolds into contact 5-manifolds. In this
talk I will discuss joint work with Ryo Furukawa aimed at using braiding
techniques to study contact embeddings. Braided embeddings give an
explicit way to represent some (maybe all) smooth embeddings and should
be useful in computing various invariants. If time permits I will also
discuss other methods for embedding and constructions one may perform on
contact submanifolds.
Dan Cristofaro-Gardiner: "Beyond the Weinstein conjecture"
Abstract: The Weinstein conjecture states that any Reeb vector field
on a closed manifold has at least one closed orbit. The
three-dimensional case of this conjecture was proved by Taubes in 2007,
and Hutchings and I later showed that in this case there are always at
least 2 orbits. While examples exist with exactly two orbits, one
expects that this lower bound can be significantly improved with
additional assumptions. For example, a theorem of Hofer, Wysocki, and
Zehnder states that a generic nondegenerate Reeb vector field associated
to the standard contact structure on $S^3$ has either 2, or infinitely
many, closed orbits. We prove that any nondegenerate Reeb vector field
has 2 or infinitely many closed orbits as long as the associated contact
structure has torsion first Chern class. This is joint work with Mike
Hutchings and Dan Pomerleano.

Series: Geometry Topology Seminar

In a recent conjecture by Tian Yang and Qingtao Chen, it has been observedthat the log of Turaev-Viro invariants of 3-manifolds at a special root ofunity grow proportionnally to the level times hyperbolic volume of themanifold, as in the usual volume conjecture for the colored Jonespolynomial.In the case of link complements, we derive a formula to expressTuraev-Viro invariants as a sum of values of colored Jones polynomial, andget a proof of Yang and Chen's conjecture for a few link complements. Theformula also raises new questions about the asymptotics of colored Jonespolynomials. Joint with Effie Kalfagianni and Tian Yang.

Series: Geometry Topology Seminar

The point-pushing subgroup of the mapping class group of a surface with a marked point can be considered topologically as the subgroup
that pushes the marked point about loops in the surface. Birman demonstrated that this subgroup is abstractly isomorphic to the fundamental
group of the surface, \pi_1(S). We can characterize this point-pushing subgroup algebraically as the only normal subgroup inside of the mapping
class group isomorphic to \pi_1(S). This uniqueness allows us to recover a description of the outer automorphism group of the mapping class group.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Dehn surgery is a fundamental tool for constructing oriented 3-Manifolds. If we fix a knot K in an oriented 3-manifold Y and do surgeries with distinct slopes r and s, we can ask under which conditions the resulting oriented manifold Y(r) and Y(s) might be orientation preserving homeomorphic. The cosmetic surgery conjecture state that if the knot exterior is boundary irreducible then this can't happen. My talk will be about the case where Y is an homology sphere and K is an hyperbolic knot.

Series: Geometry Topology Seminar

The Grothendieck group K_0 of a commutative ring is well-known to be a \lambda-ring: although the exterior powers are non-additive, they induce maps on K_0 satisfying various universal identities. The \lambda-operations are known to give homomorphisms on higher K-groups. In joint work in progress with Barwick, Glasman, and Nikolaus, we give a general framework for such operations. Namely, we show that the K-theory space is naturally functorial with respect to polynomial functors, and describe a universal property of the extended K-theory functor. This extends an earlier algebraic result of Dold for K_0.

Series: Geometry Topology Seminar

The Drinfeld double of a finite dimensional Hopf algebra is a
quasi-triangular Hopf algebra with the canonical element as the universal R
matrix, and we obtain a ribbon Hopf algebra by adding the ribbon element.
The universal quantum invariant is an invariant of framed links, and is
constructed diagrammatically using a ribbon Hopf algebra. In that
construction, a copy of the universal R matrix is attached to each positive
crossing, and invariance under the Reidemeister III move is shown by the
quantum Yang-Baxter equation of the universal R matrix.
On the other hand, R. Kashaev showed that the Heisenberg double has the
canonical element (the universal S matrix) satisfying the pentagon
relation. In this talk we reconstruct the universal quantum invariant using
Heisenberg double, and extend it to an invariant of colored ideal
triangulations of the complement. In this construction, a copy of the
universal S matrix is attached to each tetrahedron and the invariance under
the colored Pachner (2,3) move is shown by the pentagon equation of the
universal S matrix