Résumés-knots

In 1981 Gordon introduced the notion of ribbon concordance of knots and conjectured that this relation is a partial order on knots. Gordon asked several other questions in his paper which remain open, such as that the partial order is well-founded and that simplicial volume is monotonic. We will connect some of his questions to questions about representations of knot groups, and give a bit of evidence for positive answers to these questions. 

We will survey some recent progress on diffeomorphisms of 4-manifolds including how pseudo-isotopy theory can be used to approach long-standing problems. We will discuss work with Ryan Budney and recent work with David Gay and Daniel Hartman.

This talk will survey some of the major progress and challenges concerning Dehn surgery, with an emphasis on Cameron Gordon’s decisive contributions and influence.

In this talk, I will share some of my joint work with Cameron Gordon and Steve Boyer on the left-orderability of fundamental groups of 3-manifolds. This includes results concerning 3-manifolds obtained by Dehn surgery and by taking branched covers along knots.

I will discuss a new sufficient condition for when a braided link, a braid closure together with its braid axis, is bi-orderable meaning the fundamental group of its exterior admits an order invariant under both left and right multiplication. In 2006, Perron and Rolfsen provided a condition which ensures that an automorphism of a free group preserves a bi-order of the free group and shows that many fibered 3-manifolds are bi-orderable, including many exteriors of braided links. Recent work of Khanh Le and I provide another new criterion, via the Burau representation, for a free group automorphism to be order-preserving. Using the new criterion, we produce new examples of bi-orderable braided link groups including some examples produced from braids whose underlying permutation is a full cycle which answers in affirmative a question of Kin and Rolfsen. This work is partially supported by the NSF grant DMS-2213213.

I will discuss recent work with C. Leininger in which we produce purely pseudo-Anosov surface subgroups of mapping class groups, obtaining the first compact atoroidal surface bundles over surfaces. We do this by constructing a type-preserving representation of the figure eight knot group into the mapping class group of the thrice-punctured sphere.

This famous Gordon-Luecke knot complement theorem says that knots in the three-sphere are determined by their complements.  This was settled by proving that non-trivial knots in the three-sphere cannot have non-trivial Dehn surgeries to the three-sphere.  More generally, the Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds.  We discuss some results towards cosmetic surgery and knot complement problems.  This is joint work with Ali Daemi and Mike Miller Eismeier.

Dehn surgery (with fixed slope $p/q$) can be thought of as a function from the set of knots in $S^3$ to the set of closed oriented 3-manifolds with first homology $\mathbb/p$. In his 1978 survey, Some aspects of classical knot theory, Gordon conjectured that this function is never surjective or injective. Gordon-Luecke proved the surjectivity assertion in 1989. In this talk I will describe joint work with Hayden and Wakelin proving the injectivity assertion. I will also discuss some refined surjectivity and injectivity questions. 

I will survey Cameron’s contributions to the theory of knot concordance, in particular the huge impact of the Casson-Gordon invariants on the field.

I’ll give a retrospective survey of Cameron’s work on the L-space conjecture.  Since it was first stated 15 years ago, Boyer, Gordon, and Watson’s bold conjecture that prime 3-manifolds are L-spaces if and only if they have non-left-orderable fundamental groups has inspired many new ideas, and much effort spent looking for counterexamples.

The cosmetic surgery conjecture for knots in S^3 states that any two different Dehn surgeries on a nontrivial knot in S^3 are not orientation-preservingly homeomorphic. Significant recent progress towards the conjecture has been made using Floer homology, especially by Ni-Wu, Hanselman, and Daemi-Eismeier-Lidman. In this talk, we build on recent results to show that if there exists a satellite knot counterexample to the cosmetic surgery conjecture, then there exists infinitely many hyperbolic knot counterexamples. The proof is purely topological and relies on a careful analysis of the JSJ decomposition of the surgeries.

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 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. We build on work of Cai-Clay-Rolfsen to produce a new sufficient condition for a braided link group to be bi-orderable which involves linking numbers and the Burau representation of a related braid.

A rational slope p/q is characterising for a knot K if the oriented homeomorphism type of the 3-manifold obtained via p/q-surgery on K uniquely determines the knot K. I will give an overview of the progress made towards determining the set of characterising slopes for a knot, including joint work with Patricia Sorya in which we show that the JSJ decomposition of the knot exterior determines an explicit denominator bound which guarantees that a slope is characterising. 

I will outline a proof of mutation invariance for reduced Khovanov homology over any field. When restricting to mod 2 coefficients, this result can be accessed in a number of different ways—the first proofs were given by Bloom and by Wehrli. Managing signs will appeal to a geography result for the tangle-version of the Khovanov invariant, which can be stated in terms of immersed curves. Establishing this makes an interesting appeal to homological mirror symmetry for the 3-punctured sphere. This is part of a joint project with Kotelskiy and Zibrowius. 

I will survey some of Cameron Gordon’s work in infinite group theory. In particular, I will explain the influence of his work on current research themes and programmes.

A classical result of Casson and Gordon asserts that a fibered knot K is homotopy-ribbon in a homotopy 4-ball if and only if its monodromy extends over a handlebody.  This lovely theorem has influenced my work in several ways I’ll discuss in this talk.  First, in joint work with Maggie Miller, we extend this theorem to the setting of non-fibered ribbon knots.  Second, in joint work with Jeff Meier, we determine all possible handlebody extensions of the monodromy of a generalized square knot T(p,q) # -T(p,q), and we prove that T(p,2) # -T(p,2) bounds infinitely many ribbon disks up to isotopy, all of which have diffeomorphic exteriors.