slice-ribbon for links (surely it's wrong) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:29:56Z http://mathoverflow.net/feeds/question/11713 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11713/slice-ribbon-for-links-surely-its-wrong slice-ribbon for links (surely it's wrong) Ryan Budney 2010-01-14T01:13:55Z 2010-04-08T14:21:51Z <p>The slice-ribbon conjecture asserts that all slice knots are ribbon. </p> <p>This assumes the context: </p> <p>1) A `knot' is a smooth embedding $S^1 \to S^3$. We're thinking of the 3-sphere as the boundary of the 4-ball $S^3 = \partial D^4$. </p> <p>2) A knot being slice means that it's the boundary of a 2-disc smoothly embedded in $D^4$. </p> <p>3) A slice disc being ribbon is a more fussy definition -- a slice disc is in ribbon position if the distance function $d(p) = |p|^2$ is Morse on the slice disc and having no local maxima. A slice knot is a ribbon knot if one of its slice discs has a ribbon position.</p> <p>My question is this. All the above definitions have natural generalizations to links in $S^3$. You can talk about a link being slice if it's the boundary of disjointly embedded discs in $D^4$. Similarly, the above ribbon definition makes sense for slice links. Are there simple examples of $n$-component links with $n \geq 2$ that are slice but not ribbon? Presumably this question has been investigated in the literature, but I haven't come across it. Standard references like Kawauchi don't mention this problem (as far as I can tell). </p> http://mathoverflow.net/questions/11713/slice-ribbon-for-links-surely-its-wrong/12325#12325 Answer by czy for slice-ribbon for links (surely it's wrong) czy 2010-01-19T17:56:39Z 2010-01-19T17:56:39Z <p>As far as I know, extension of slice and ribbon to links is not unique. There are "strong slice", "weak slice", "strong ribbon" and "weak ribbon" for links.</p> <p>"CHARACTERIZATION OF SLICES AND RIBBONS" (by H.FOX) mentioned these concepts. </p> http://mathoverflow.net/questions/11713/slice-ribbon-for-links-surely-its-wrong/18971#18971 Answer by Peter Teichner for slice-ribbon for links (surely it's wrong) Peter Teichner 2010-03-22T01:38:51Z 2010-03-22T01:38:51Z <p>Ryan, I think this is an open problem. The best related result I know is a theorem of Casson and Gordon [A loop theorem for duality spaces and fibred ribbon knots. Invent. Math. 74 (1983)] saying that for a fibred knot that bounds a homotopically ribbon disk in the 4-ball, the slice complement is also fibred. </p> <p>More precisely, they are assuming that the knot K bounds a disk R in the 4-ball such that the inclusion </p> <p>$S^3 \smallsetminus K \hookrightarrow D^4 \smallsetminus R$ </p> <p>induces an epimorphism on fundamental groups. If one glues R to a fibre of the fibration $S^3 \smallsetminus K \to S^1$ to obtain a closed surface F, then the statement is that the monodromy extends from F to a solid handlebody which is a fibre of a fibration $D^4 \smallsetminus R \to S^1$ extending the given one on the boundary.</p>