In mathematics, the slice genus of a smooth knot ''K'' in ''S''3 (sometimes called its Murasugi genus or 4-ball genus) is the least integer g such that ''K'' is the boundary of a connected, orientable 2-manifold ''S'' of genus ''g'' embedded in the 4-ball ''D''4 bounded by ''S''3.
More precisely, if ''S'' is required to be smoothly embedded, then this integer ''g'' is the ''smooth slice genus'' of ''K'' and is often denoted gs(''K'') or g4(''K''), whereas if ''S'' is required only to be topologically locally flatly embedded then ''g'' is the ''topologically locally flat slice genus'' of ''K''. (There is no point considering ''g'' if ''S'' is required only to be a topological embedding, since the cone on ''K'' is a 2-disk with genus 0.) There can be an arbitrarily great difference between the smooth and the topologically locally flat slice genus of a knot; a theorem of Michael Freedman says that if the Alexander polynomial of ''K'' is 1, then the topologically locally flat slice genus of ''K'' is 0, but it can be proved in many ways (originally with gauge theory) that for every g there exist knots ''K'' such that the Alexander polynomial of ''K'' is 1 while the genus and the smooth slice genus of ''K'' both equal g.
The (smooth) slice genus of a knot ''K'' is bounded below by a quantity involving the Thurston–Bennequin invariant of ''K'':
The (smooth) slice genus is zero if and only if the knot is concordant to the unknot.
*Milnor conjecture (topology)
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