Isotopy and equivalence of knots in 3-manifolds

TL;DR

This paper investigates conditions under which knots in prime, closed, oriented 3-manifolds are isotopic, linking this to the triviality of the mapping class group, and explores knot isotopy behavior under specific homeomorphisms.

Contribution

It establishes a criterion relating knot isotopy to the triviality of the mapping class group in prime 3-manifolds and analyzes isotopy classes under Gluck twists in S^1×S^2.

Findings

Equivalent knots are isotopic iff the mapping class group is trivial.

Homeomorphisms preserving free homotopy classes are isotopic to the identity in certain manifolds.

Gluck twists can change knot isotopy classes in S^1×S^2.

Abstract

We show that in a prime, closed, oriented 3-manifold M, equivalent knots are isotopic if and only if the orientation preserving mapping class group is trivial. In the case of irreducible, closed, oriented $3$-manifolds we show the more general fact that every orientation preserving homeomorphism which preserves free homotopy classes of loops is isotopic to the identity. In the case of $S^1\times S^2$, we give infinitely many examples of knots whose isotopy classes are changed by the Gluck twist.