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This talk will introduce some of the basic tools of orderable groups, such as the Burns-Hale theorem, the space of orderings, and the behaviour of orderability with respect to standard group-theoretic constructions. I'll also touch on how these are used in low-dimensional topology, and briefly mention a few open questions along the way.
The L-space conjecture asserts the equivalence of three properties of a prime 3-manifold $M$: (1) the fundamental group of $M$ has a left-invariant total order, (2) $M$ admits a taut 2-dimensional foliation, and (3) the Heegaard Floer homology of $M$ is "large". One of the interests of this conjecture is that these properties have very different flavors: (1) is algebraic, (2) is topological, and (3) is essentially analytic.
We will define the three properties and describe some things that are known about the conjecture. In particular we will discuss some recent joint work with Steve Boyer and Ying Hu.
In this talk, I will introduce the notion of orderability of quandles (right and left orderability, right and left circularly orderability, and bi-orderability of quandles), and its relation with the notion of orderability of 3-manifold groups. I will also give new techniques to study the bi-orderability of knot groups by using quandles and also give examples (this is a joint work with Mohamed Elhamdadi).
The mapping class group of a surface with a single boundary component admits a natural action on the real line by orientation-preserving homeomorphisms. Capping off the boundary component, the mapping class group of a surface with one marked point admits a natural action on the circle $S^1$, with the action on $S^1$ related to the action on $\mathbf{R}$ by a quotienting/lifting argument. In this talk I'll discuss how the rigidity of these actions allows us to compute fractional Dehn twist coefficients from left-orderings of mapping class groups, including recent results in the low-genus cases. This is joint work with Ty Ghaswala and Idrissa Ba.
Motivated by an observation of Dehornoy, we study the roots of Alexander polynomials of knots and links that are closures of positive 3-strand braids. We give experimental data on random such braids and find that the roots exhibit marked patterns, which we refine into precise conjectures. We then prove several results along those lines, for example that generically at least 69% of the roots are on the unit circle, which appears to be sharp. We also show there is a large root-free region near the origin. We further study the equidistribution properties of such roots by introducing a Lyapunov exponent of the Burau representation of random positive braids, and a corresponding bifurcation measure. In the spirit of Deroin and Dujardin, we conjecture that the bifurcation measure gives the limiting measure for such roots, and prove this on a region with positive limiting mass. We use tools including work of Gambaudo and Ghys on the signature function of links, for which we prove a central limit theorem. This is joint work with Giulio Tiozzo.
The concept of slope detections was originally introduced by Boyer and Clay to study three conjecturally equivalent properties for toroidal manifolds: not being an L-space (NLS), having left-orderable fundamental group (LO), and admitting a co-orientable taut foliation (CTF). In this talk, we emphasize the importance of detecting meridional slopes of knots in integer homology spheres. We show that they are detected by the properties LO and NLS and also by CTF if the knot manifolds are fibered. We show applications of meridional detection to toroidal rational homology spheres with small first homology, and Dehn surgeries on satellite knots. This is joint work with Steve Boyer and Cameron Gordon.
A group is bi-orderable if it admits a total ordering that is left and right invariant. Given a bi-orderable group $G$, it is natural to ask which outer automorphisms of $G$ preserve a bi-ordering on $G$ since these correspond precisely to cyclic extension of $G$ that is bi-orderable. In this talk, I will give a complete characterization of finite-order outer automorphisms of non-abelian free and surface groups that are order-preserving. I will explain how this result directly generalizes the work of Kin and Rolfsen on order-preserving periodic braids and is related to the classification of bi-orderable Seifert fibred space by Boyer, Rolfsen and Wiest. This is a work in progress joint with Jonathan Johnson.
Let $M$ be a closed, orientable, and irreducible 3-manifold with Heegaard genus two. We prove that if the fundamental group of $M$ is left-orderable then $M$ admits a co-orientable taut foliation.
Codimension-one foliations of 3-manifolds will be defined and discussed. Then smoothness of foliations will become the focus and I will discuss a result, joint with John Cantwell and Larry Conlon, about the smoothness of Gabai's finite depth foliations.
Any link (or knot) group – the fundamental group of a link complement – is left-orderable. However, not many link groups are bi-orderable – that is, admit an order invariant under both left and right multiplication. It is not well understood which link groups are bi-orderable, nor is there is a conjectured topological characterization of links with bi-orderable link groups. I will discuss joint work in progress with Jonathan Johnson and Nancy Scherich to study this problem for braided links – braid closures together with their braid axis. Inspired by Kin-Rolfsen, we focus on braided link groups because algebraic properties of the braid group can be employed in this setting. In particular, I will discuss our implementation of an algorithm which, given a braided link group which is not bi-orderable, will return a definitive "no" and a proof in finite time. Using our program, we give a new infinite family of non-bi-orderable braided links. This is joint work in progress with Jonathan Johnson and Nancy Scherich.
For a braid-type automorphism $\varphi$ on a free group $F$, we give a sufficient condition for the existence of a bi-ordering of $F$ invariant under the $\varphi$. As an application, it immediately follows that the fundamental group of magic manifold $S_3-\mathrm{br}(\sigma_1^2\sigma_2^{-1})$ is bi-orderable, thus solving a problem of Kin-Rolfsen.
This is a joint work with Adam Clay and Dale Rolfsen.
The notion of slope detection was first introduced by Boyer and Clay to study the L-space conjecture for graph manifolds and has then been developed further by many others under the context of the L-space conjecture. After developing a general definition of order-detection of slopes on a 3-manifold with multiple incompressible torus boundary components and new relative gluing theorems, we are able to describe order-detected slopes on cable knots, whose behavior is similar to the results by Hedden and Hom concerning the behavior of L-space knots with respect to cabling.
Boyer-Gordon-Hu introduced slope detections to study L-space conjecture for toroidal manifolds. For a 3-manifold $M$ with a torus boundary, a boundary slope is called CTF-detected if there’s a co-oriented taut foliation of $M$ intersecting the boundary torus transversely in a suspension foliation of the slope. We give a highly inductive proof to show that for any (1,1)-knot complement, the meridional slope (with respect to the knot) is CTF-detected.
In this talk we consider orientable 3-manifolds that arise from Dehn surgery along Montesinos knots. We will discuss the classification of Montesinos knots described through continued fractions, and study their Seifert surfaces of minimal genus. We then use this information to find persistent foliations with the help of two key lemmas.
Here is my abstract. We show that: if a closed orientable irreducible 3-manifold admits a co-orientable taut foliation with orderable cataclysm, then it has left orderable fundamental group. This provides an elementary proof that if a 3-manifold admits an Anosov flow with a co-orientable stable foliation, then it has left orderable fundamental group, which does not rely on Thurston’s universal circle action. Also, for every closed orientable 3-manifold that admits a pseudo-Anosov flow $X$ with a co-orientable stable foliation, our result applies to infinitely many of Dehn fillings along the union of singular orbits of $X$.