About: Slice genus

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In mathematics, the slice genus of a smooth knot K in S3 (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 properly embedded in the 4-ball D4 bounded by S3. 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.

Attributes	Values
rdfs:label	Slice genus (en)
rdfs:comment	In mathematics, the slice genus of a smooth knot K in S3 (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 properly embedded in the 4-ball D4 bounded by S3. 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. (en)
dcterms:subject	Knot theory
Wikipage page ID	5418032 (xsd:integer)
Wikipage revision ID	1068856758 (xsd:integer)
Link from a Wikipage to another Wikipage	Mathematics Gauge theory Elsevier Milnor conjecture (topology) Proceedings of the American Mathematical Society Link concordance Local flatness Alexander polynomial Knot genus Knot theory Knot theory Michael Freedman Slice knot Unknot Thurston–Bennequin number
sameAs	Slice genus Slice genus Slice genus
dbp:wikiPageUsesTemplate	dbt:Cite_journal dbt:Short_description dbt:Isbn dbt:MathSciNet dbt:Knottheory-stub
has abstract	In mathematics, the slice genus of a smooth knot K in S3 (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 properly embedded in the 4-ball D4 bounded by S3. 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. (en)
prov:wasDerivedFrom	wikipedia-en:Slice_genus?oldid=1068856758&ns=0
page length (characters) of wiki page	2495 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Slice_genus
is Link from a Wikipage to another Wikipage of	List of knot theory topics Milnor conjecture (topology) Khovanov homology Knot invariant Slice Slice knot Seifert surface
is Wikipage disambiguates of	Slice
is foaf:primaryTopic of	wikipedia-en:Slice_genus

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