Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

TBA

Symmetric unions are an interesting class of knots. Although they have not been studied much for their own sake, they frequently appear in the literature. One such instance regards the question of whether there is a nontrivial knot with trivial Jones polynomial. In my talk, I will describe a class of symmetric unions, constructed by Tanaka, such that if any are amphichiral, they would have trivial Jones polynomial. Then I will show how such a knot not only answers the above question but also gives rise to a counterexample to the Cosmetic Surgery Conjecture. However, I will prove that such a knot is in fact trivial and hence cannot be used to answer any of these questions. Finally, I will discuss how the arguments that go into this proof can be generalized to study amphichiral symmetric unions.

Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from algebraic surfaces and curves inside them. Némethi created lattice homology as an invariant for links of normal surface singularities which developed out of computations for Heegaard Floer homology. Later Ozsváth, Stipsicz, and Szabó defined knot lattice homology for generalized algebraic knots in rational homology spheres, which is known to play a similar role to knot Floer homology and is known to compute knot Floer in some cases. I discuss a proof that knot lattice is an invariant of the smooth knot type, which had been previously suspected but not confirmed.