Isotopy and equivalence of knots in 3‐manifolds

Source abstract

Abstract Two knots and in are isotopic if and only if they are related by an orientation‐preserving diffeomorphism of . This claim follows from the fact that any orientation‐preserving self‐diffeomorphism of is isotopic to the identity. We show that this same idea applies to any prime oriented closed 3‐manifold. More precisely, we show that a prime closed oriented 3‐manifold contains a pair of equivalent but nonisotopic knots if and only if the (orientation‐preserving) mapping class group is nontrivial. When is additionally irreducible we show that an orientation‐preserving diffeomorphism of is isotopic to the identity if and only if it preserves all homotopy classes of knots. For knots in (the only reducible prime oriented 3‐manifold) we exhibit infinitely many knots whose isotopy classes are not preserved by the Gluck twist.