User kave - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:01:26Z http://mathoverflow.net/feeds/user/23155 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94974/slice-knots-what-does-the-locally-flat-condition-say slice knots, what does the locally flat condition say? kave 2012-04-23T19:54:19Z 2012-04-23T23:44:29Z <p>I am studying slice knots, so for example they say the cone on a trefoil knot can be embedded in D^4 but it is not locally flat at the vertex of the cone. What I do not understand that I think every surface which is embedded in D^4 is already flat. What prevents the cone to be flat at the vertex? after all in D^4 we have enough room to slide the surface over its intersections it may have in three dimension and so it does not have any intersections with itself, doesn't this mean that it is flat?(if we want to think about smoothness then it seems that even if it is not smooth at some point then we can make it smooth at that point easily). I maybe wrong about the ways a surface can be non-smooth. So one more question can you give me an example in lower dimensions, so that for example a curve which can be embedded in R^3 but it is not flat (or non smooth, say)?</p>