##### Department of Mathematics,

University of California San Diego

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### Topology Seminar

## Matthew Hedden

#### Massachusetts Institute of Technology

## On knot Floer homology and complex curves

##### Abstract:

\noindent Suppose we view the three-dimensional sphere as: $S^3 = \{(z,w) \subset \mathbb{C}^2|\ |z|^2 + |w|^2 = 1\}. $ If we are given a complex curve $V_f = \{(z,w)|0 = f(z,w) \in \mathbb{C} [z,w]\},$ we can then examine the intersection $K = V_f \cap S^3.$ In the transverse case, this intersection $K$ will be a link i.e. an embedded one-manifold in the three-sphere. This talk will be interested in the question: Question: Which links can arise from complex curves in the above manner? I will discuss the history of this problem, focusing first on the case where $f(z,w)$ has an isolated singularity at the origin where the question is completely answered. I’ll then discuss how a powerful set of knot invariants defined by Ozsvath and Szabo and independently by Rasmussen using the theory of pseudo-holomorphic curves can provide information on the above question. More precisely, Ozsvath and Szabo and Rasmussen defined a numerical invariant of knots, denoted $\tau(K)$, which we show provides an obstruction to knots arising in the above manner. More surprisingly, suppose we focus on knots whose exteriors, $S^3 - K$, admit the structure of a fiber bundle over the circle, the so-called $fibered$ knots. In this case we show that $\tau(K)$ detects exactly when a fibered knot arises as the intersection of the three-sphere with a complex curve satisfying a certain genus constraint. Our proof relies on connections between Ozsvath-Szabo theory and certain geometric structures on three-manifolds called contact structures.

Host: Nitya Kitchloo

### February 16, 2007

### 2:00 PM

### AP&M 6402

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