The $4$-genus of a knot is an important measure of complexity, related tothe unknotting number. A fundamental result used to study the $4$-genusand related invariants of homology classes is the Thom conjecture,proved by Kronheimer-Mrowka, and its symplectic extension due toOzsvath-Szabo, which say that closed symplectic surfacesminimize genus.Suppose (X, \omega) is a symplectic 4-manifold with contact type bounday and Sigma is a symplectic surface in X such that its boundary is a transverse knot in the boundary of X.
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