Georgia Institute of Technology College of Sciences

Anosov flows provide beautiful examples of interactions between dynamics, geometry and analysis. In dimension 3 in particular, they are known to have a subtle relation to topology as well. Motivated by a result of Mitsumatsu from 1995, I will discuss their relation to contact and symplectic structures and argue why contact topological methods are natural tools to study the related global phenomena.

The emergent shape of a knitted fabric is highly sensitive to the underlying stitch pattern. Here, by a stitch pattern we mean a periodic array of symbols encoding a set of rules or instructions performed to produce a swatch or a piece of fabric. So, it is crucial to understand what exactly these instructions mean in terms of mechanical moves performed using a yarn (a smooth piece of string) and a set of knitting needles (oriented sticks). Motivated by the fact that locally every knitting move results in a slip knot, we use tools from topology to model the set of all doubly periodic stitch patterns, knittable & non-knittable, as knots & links in a three manifold. Specifically, we define a map from the set of doubly-periodic stitch patterns to the set of links in S^3 and use link invariants such as the linking number, multivariable Alexander polynomial etc. to characterize them. We focus on such links derived from knitted stitch patterns in an attempt to tackle the question: whether or not a given stitch pattern can be realized through knitting.

Which manifold can be obtained from surgery on a knot? Many obstructions to this have been studied. We will discuss some of them, and use Heegaard Floer homology to give an infinite family of seifert fibered integer spheres that cannot be obtained by surgery on a knot in S^3. We will also discuss a recipe to compute HF+ of surgery on a knot (Short review on Heegaard Floer homology included).