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The 4-genus of a knot is an important measure of complexity, related to the unknot-ting number. A fundamental result used to study the 4-genus and related invariants
of homology classes is the Thom conjecture, proved by Kronheimer-Mrowka, and its symplectic extension due to Ozsvath-Szabo, which say that closed symplectic surfaces
minimize genus.
Suppose (X, ω) is a symplectic 4-manifold with contact type bounday ∂X and Σ is a symplectic surface in X such that ∂Σ is a transverse knot in ∂X. In this talk we show that there is a closed symplectic 4-manifold Y with a closed symplectic submanifold S such that the pair (X, Σ) embeds symplectically into (Y, S). This gives a proof of the relative version of Symplectic Thom Conjecture. We use this to study 4-genus of fibered knots in S^3.
We will also discuss a relative version of Giroux’s criterion for Stein fillability.
This is joint work with Siddhartha Gadgil. |