Résumés-knots

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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.

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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 $\mathbb{Z}/p$. 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.

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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.

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