The $4$-genus of a knot is an important measure of complexity, related to the unknotting number. A fundamental result used to study the $4$-genus and related invariants of homology classes is the \emph{Thom Conjecture}, proved by Kronheimer-Mrowka, and its symplectic extension due to Ozsvath-Szabo, which say that \textit{closed} symplectic surfaces minimize genus.

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Last updated: 11 Dec 2023