Expository paper on the construction of the Alexander polynomial. Covers basic knot theory, and the fundamental group of the knot complement.

Essay on a result of Levine and Lidman on the existence of smooth, simply-connected, spineless 4-manifolds. The existence of simply-connected, spineless 4-manifolds appeared as problem 4.25 in Kirby’s problem list in the 1970s, and remained open until 2018 when it was closed by Levine and Lidman. Essay covers relevant background material ($\text{spin}^c$ structures, Heegaard splittings, Kirby calculus and knot surgery, Heegaard Floer Homology), describes the constructions of such spineless manifolds and explores associated literature.

Manuscript serves on Milnor's exotic 7-spheres. The exotica served as a counterexample to the differentiable Poincaré conjecture. The discovery of exotic 7-spheres also led to a revolution in the newly blossoming field of differential geometry, and envigorated a race to discover exotic structures in other dimensions. In 4 dimensions, the problem still remains open. Manuscript contains background material on principal bundles, homology and cohomology theory, characteristic classes, and foundational Morse theory.

Abstract: I will talk about a result of Hutchings, 2011 proving the following result, previously a conjecture of Arnold, 1986: Theorem: Let Y be a closed oriented 3-manifold with a contact form λ. Then every Legendrian knot in (Y,λ) has a Reeb chord. The proof makes use of Legendrian knot surgery, the change in Reeb dynamics under such a procedure, and the maps on Embedded Contact Homology induced by the surgery cobordisms. In particular, the proof borrows an important result from cobordism maps in Seiberg-Witten Floer Cohomology to guarantee the existence of Reeb chords. I will outline the description of Legendrian knot surgery used by Hutchings and explore how one may force the existence of Reeb chords to appear by introducing the natural filtration on ECH and the induced cobordism maps.

Abstract: The Jones polynomial’s origins lie in a purely algebraic setting, first described by Vaughn Jones via representations of the Braid group into the Temperley-Lieb algebra. As any knot has a presentation in the braid group, this allows us to associate to any knot a Jones polynomial. One can reformulate the Jones polynomial in terms of resolutions of knot diagrams, from which one can explicitly show that it is a knot invariant. We now shift the discussion away from constructions of the Jones polynomial, and to one of its classical applications – detecting the unknot. Identifying one knot from another is in general a very hard problem, in fact, one that has only found some resolution in recent decades. Certain polynomial invariants like the Alexander polynomial can make some detections but fail in general. An open problem is whether the Jones polynomial can do so. As of yet, there is no answer to the conjecture If K is a knot, then V(K)=1 if and only if K=U. In this talk, we’ll investigate unknotting in two parts. First, we’ll discuss the significance and depth of the unknotting problem, and survey the methods of unknot detection we have thus far. Second, we’ll discuss the potential of the Jones polynomial as a tool for unknot detection and probe the possibility of extending positive results for certain types of knots to the more general case.

Abstract: A PL spine of a manifold is a PL-embedded submanifold of strictly lower dimension, which is also a strong deformation retract. The question of the existence of spines in manifolds has persisted since the 1960s, with interesting results emerging for those of codimension 2. The work of Cappell and Shaneson in 1976 closed the case of codimension-2 spines in manifolds of dimension greater than 4 (1976), and in 1975 Matsumoto proved the existence of a noncompact manifold that does not admit a codimension-2 PL spine in dimension 4. Matsumoto also conjectured the existence of a compact 4-manifold that does not admit a PL spine; this problem appeared later as Problem 4.25 in Kirby’s problem list. A result by Levine and Lidman in 2018 finally proved the conjecture. Specifically, Levine and Lidman showed that there exist infinitely many smooth, compact, simply-connected 4-manifolds that are homotopy equivalent to the 2-sphere but do not admit a PL embedding realizing this homotopy equivalence. In this talk, we will give a detailed overview of their paper constructing the candidate spineless 4-manifolds via Kirby calculus, and defining an obstruction to a spine via the d-invariant arising from Heegaard Floer homology. Time permitting, we will discuss alternative proofs closing problem 4.25 of Kirby's list, and the broader application of invariants coming from Heegaard Floer homology in exploring exotic phenomena in dimension 4.

Abstract: The question of whether the Conway knot bounds a smoothly embedded disc in the 4-ball (i.e. slice) stumped knot theorists for over 50 years until Lisa Piccirillo solved the problem in a mere 6-page paper. Her proof makes use of RGB-diagrams, and Rasmussen's s-invariant arising from Khovanov homology. This talk will unpack the details of Piccirillo's argument, focusing on the construction of an auxiliary knot whose sliceness is equivalent to sliceness of the Conway knot. We will also describe the computation of the s-invariant of this auxiliary knot, which will allow us to conclude that it, and therefore the Conway knot, is not slice.

Abstract: We continue our previous investigation into the construction of Heegaard Floer Homology from part I of this talk. The first part of the talk covered the construction of Lagrangian Floer Homology (LFH). LFH gives an invariant of a pair of Lagrangian submanifolds embedded in a symplectic manifold. We continue our exploration by decomposing a 3-manifold into its essential parts by specifying a Heegaard splitting. A way of representing this data, a Heegaard diagram, provides all the information we need in order to construct a symplectic manifold and lagrangian submanifold pair to run through the LFH machine. Given sufficient time, we will translate this construction into some concrete examples, and calculate the (hat) HFH of the 3-sphere. We'll make some small remarks on the proof of this result, and highlight properties of HFH that makes it so useful.

Abstract: Heegaard Floer homology associates a package of invariants to a 3-manifold, which are constructed in an analogous fashion to Lagrangian Floer homology and isomorphic to Seiberg-Witten homology. This talk aims to describe the very basic motivation and construction of HFH restricted to the case of homology 3-spheres. We will spend time discussing the context in which Heegaard Floer arose, and create a roadmap to understand the mechanics of these homology groups. A consecutive talk will focus on justifying why these constructions are in fact topological invariants, and will describe some of their useful properties.