Georgia Institute of Technology College of Sciences

Dehn surgery (with a fixed slope p/q) associates, to a knot in S^3, a 3-manifold M with first homology isomorphic to the integers mod p. One might wonder if this function is one-to-one or onto; Cameron Gordon (1978) conjectured that it is never injective nor surjective. The surjectivity case was established a decade later, while the injectivity case was only recently proven by Hayden, Piccirillo, and Wakelin. We will survey this latter result and its proof. 

Taking the double branched cover of $S^3$ over a knot $K$ is natural way to associate $K$ with a 3-manifold, and to study the double branched cover, we often want a Dehn surgery description for it. The Montesinos trick gives a systematic way to get such a description. In this talk, we will go over the broad statement of this trick: that a rational tangle replacement on the knot corresponds to Dehn surgery on the double branched cover. This gives particularly nice descriptions for some satellites of $K$ as surgery on $K \mathrel\# K^r$. We will also discuss an application of the trick which characterizes the 2-bridge knots with unknotting number 1.

The Fox trapezoidal conjecture is a longstanding open problem about the coefficients of the Alexander polynomial of alternating links. In this talk, we will discuss recent work which settled this conjecture for “special alternating links”. The first tool is a graph theoretic model of the Alexander polynomial of an alternating link discovered by Crowell in 1959. The second is the theory of Lorentzian polynomials, developed by Brändén and Huh in 2019 and a key part of Huh’s Fields medal work. We will show how a version of Crowell’s model produces a refinement of the Alexander polynomial of special alternating links that is Lorentzian, from which the result follows quickly.

We discuss the proof of the following Theorem

Assume $E$ is a $C^{p}$ real Banach manifold, and $f:E\circlearrowleft$, $f\circ f=f$ is a $C^{p}$ retraction, where $1\leq p\leq +\infty$. Then the retract $f(E)$ is a $C^{p}$ sub Banach manifold of $E$.

If time allows, we will also discuss how this fact is related to the study of smoothness and structural stability of attractors, along the intersection of topology and dynamics. We will be focusing on the proof and perspective of Oliva 1975, who was interested in Banach manifolds as phase-spaces of delay equations.