词条 | Slice genus |
释义 |
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. See also
Further reading
| author = 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 3050 | year = 1997 |mr=1443854 | doi = 10.1090/S0002-9939-97-04258-5 }}
1 : Knot theory |
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