15 and 290 theorems: Difference between revisions

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m clean up using AWB (9767)
Statement improperly said integral quadratic form when the forms are actually more limited to quadratic forms with integral matrix.
 
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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&nbsp;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&nbsp;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&nbsp;<var>g</var>.
 
The (smooth) slice genus of a knot ''K'' is bounded below by a quantity involving the [[Thurston&ndash;Bennequin number|Thurston&ndash;Bennequin invariant]] of ''K'':
 
: <math> g_s(K) \ge ({\rm TB}(K)+1)/2. \, </math>
 
The (smooth) slice genus is zero if and only if the knot is [[Link concordance|concordant]] to the [[unknot]].
 
==See also==
*[[Slice knot]]
 
==Further reading==
*{{cite journal
| author = [[Lee Rudolph|Rudolph, Lee]]
| title = The slice genus and the Thurston-Bennequin invariant of a knot
| journal = [[Proceedings of the American Mathematical Society]]
| volume = 125
| issue = 10
| pages = 3049&nbsp;3050
| year = 1997
| id = {{MathSciNet | id = 1443854 }}
| doi = 10.1090/S0002-9939-97-04258-5 }}
 
* [[Charles Livingston|Livingston, Charles]], A survey of classical knot concordance, in: ''Handbook of knot theory'', pp 319&ndash;347, [[Elsevier]], Amsterdam, 2005. {{MathSciNet | id = 2179265 }} ISBN 0-444-51452-X
 
[[Category:Knot theory]]
 
 
{{knottheory-stub}}

Latest revision as of 05:54, 16 October 2014

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