Title: Box-dot diagrams for “regular” rational tangles
Abstract: We introduce a new presentation for rational tangles which illustrates a geometric connection to the number theory of positive regular continued fractions. This presentation also admits a suitable extension to the contact setting, allowing us to define a natural Legendrian embedding of a particular class of rational tangles into the standard contact Euclidean 3-space. We will briefly discuss how these box-dot diagrams, along with an associated construction, can be used to determine when the Legendrian flyping operation yields tangles which are not Legendrian isotopic.
Title: The non-orientable 4-genus of knots
Abstract: Every knot in the 3-sphere bounds an embedded orientable surface in the 4-ball. The minimum genus of such a surface defines the 4-genus of a knot. This is a well-studied invariant, closely related to more general problems in 4-manifold theory. The case of non-orientable surfaces has been much more challenging; the natural analogs of basic results in the orientable setting have yielded intractable questions in the non-orientable setting. In this talk I will first review some of the techniques and results of the orientable case. Then, I will discuss some of their extensions and progress in the non-orientable setting. The talk will also touch on the distinction between the topological and smooth settings.