Georgia Institute of Technology College of Sciences

I will talk briefly about how the study of fibred knots and Thurston's classification of automorphisms of surfaces in the 70's lead to Gabai and Oertel's work on essential laminations in the 80's. Some of this structure, for instance fractional Dehn twist coefficients, has implications in contact topology. I will describe results and examples, both old and new, that emphasize the special nature of S^3. This talk is based on joint work with Rachel Roberts.

Let S_g be a closed orientable surface of genus g > 1 and C a simple closed nonseparating curve in S_g. Let t_C denote a left handed Dehn twist about C. A fractional power of t_C of exponent L/n is a h in Mod(S_g) such that h^n = t_C^L. Unlike a root of a t_C, a fractional power h can exchange the sides of C. We will derive necessary and sufficient conditions for the existence of both side-exchanging and side-preserving fractional powers. We will give some applications of the main result in both cases. Finally, we give a complete classification of a certain class of side-preserving and side-exchanging fractional powers on S_5.

Recently, Sarkar showed that a smooth marked cobordism between two knots K_1 , K_2 induces a map between the knot Floer homology groups of the two knots HFK(K_1 ), HFK(K_2 ). It has been conjectured that this map is well defined (with respect to smooth marked cobordisms). After outlining what needs to be shown to prove this conjecture, I will present my current progress towards showing this result for the combinatorial version of HFK. Specifically, I will present a generalization of Carter and Saito's movie move theorem to grid diagrams, give a very brief introduction to combinatorial knot Floer homology and then present a couple of the required chain homotopies needed for the proof of the conjecture.

Berge has a construction that produces knots in S^3 that admit a lens space surgery. Conjecturally, his construction produces all such knots. This talk will consider knots that have such a surgery, and some of their contact geometric properties. In particular, knots in S^3 with a lens space surgery are fibered, and they all support the tight contact structure on S^3. From recent work of Hedden and Plamenevskaya, we also know that the dual to a lens space surgery on such a knot supports a tight contact structure on the resulting lens space. We consider the knots that are dual to Berge's knots, and we investigate whether the tight contact structure they support is a universally tight structure. Our results indicate a relationship between supporting this universally tight structure and being dual to a torus knot.