I want to show (although Artin gave an ad hoc proof) that if two braids \beta and \beta' are isotopic as braids, then they are equivalent as tangles. I'd like to use the homotopy extension property (HEP) to show that there is an isotopy H of the tangle ball B^3 which is the identity on the boundary of B^3 and at one end (ie. H_0(x)=x all x in B^3), and throws \beta onto \beta' at the other; ie. H_1(\beta)=\beta' (sometimes called an "ambient" isotopy). The problem is that the HEP gives me a homotopy, whereas I want an isotopy, ie. H_t is a homeomorphism B^3 --> B^3 for all t in I=[0,1].
Can someone please help?