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| In [[mathematics]], the '''slice genus''' of a smooth [[knot theory|knot]] ''K'' in ''S''<sup>3</sup> (sometimes called its '''[[Kunio Murasugi|Murasugi]] genus''' or '''4-ball genus''') is the least integer <var>g</var> such that ''K'' is the boundary of a connected, orientable 2-manifold ''S'' of genus ''g'' embedded in the 4-ball ''D''<sup>4</sup> bounded by ''S''<sup>3</sup>.
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| 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 <var>g<sub>s</sub></var>(''K'') or <var>g</var><sub>4</sub>(''K''), whereas if ''S'' is required only to be [[local flatness|topologically locally flat]]ly 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 <var>g</var> 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 <var>g</var>.
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| The (smooth) slice genus of a knot ''K'' is bounded below by a quantity involving the [[Thurston–Bennequin number|Thurston–Bennequin invariant]] of ''K'':
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| : <math> g_s(K) \ge ({\rm TB}(K)+1)/2. \, </math> | |
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| The (smooth) slice genus is zero if and only if the knot is [[Link concordance|concordant]] to the [[unknot]].
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| ==See also==
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| *[[Slice knot]]
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| ==Further reading==
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| *{{cite journal
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| | author = [[Lee Rudolph|Rudolph, Lee]]
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| | title = The slice genus and the Thurston-Bennequin invariant of a knot
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| | journal = [[Proceedings of the American Mathematical Society]]
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| | volume = 125
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| | issue = 10
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| | pages = 3049 3050
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| | year = 1997
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| | id = {{MathSciNet | id = 1443854 }}
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| | doi = 10.1090/S0002-9939-97-04258-5 }}
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| * [[Charles Livingston|Livingston, Charles]], A survey of classical knot concordance, in: ''Handbook of knot theory'', pp 319–347, [[Elsevier]], Amsterdam, 2005. {{MathSciNet | id = 2179265 }} ISBN 0-444-51452-X
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| [[Category:Knot theory]]
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| {{knottheory-stub}}
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Hello and welcome. My name is Numbers Wunder. My working day job is a librarian. One of the things he loves most is ice skating but he is having difficulties to find time for it. Years ago we moved to North Dakota and I adore every working day living here.
Check out my homepage home std test kit (simply click the up coming post)