ambient isotopy and isotopy on knot - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T02:21:26Z http://mathoverflow.net/feeds/question/37942 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37942/ambient-isotopy-and-isotopy-on-knot ambient isotopy and isotopy on knot Seonhwa Kim 2010-09-07T02:46:06Z 2010-09-07T02:46:06Z <p>this is elementary question about classical knot equivalence. </p> <p>I know that just isotopy which need not to be ambient is not proper to define knot equivalence because bachelor's unknotting.</p> <p>but this example is unsatisfied to me, because this unknotting isotopy is not differentiable.</p> <p>If the isotopy is given "smooth" condition, it seems to define knot equivalence intuitively. i.e $F(x,t): S^1 \times [0,1] \to S^3$ s.t $F(x,t)$ is smooth map and level restriction $F(x,t_0)$ is embedding. Can it define knot equivalence?</p> <p>I think it may be false.</p> <p>Any reference is grateful to me.</p>