Isotopy and equivalence of knots in three-manifolds

Abstract: It is a well-known fact that the notions of (ambient) isotopy and equivalence coincide for knots in \(S^3\). This is because all orientation-preserving homeomorphisms of \(S^3\) are isotopic to the identity. In this talk, we compare the notions of equivalence and isotopy for knots in more general three-manifolds.

We show that the mapping class group of a three-manifold "sees" all the isotopy classes of knots; that is, if an orientation-preserving homeomorphism fixes every isotopy class, then it 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. Along the way we prove that every three-manifold group satisfies Grossman's Property A.

This is joint work with Paolo Aceto, Christopher Davis, JungHwan Park, and Arunima Ray.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

( chat | slides )

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Geometry and topology online

Series comments: You can also find up-to-date information on the seminar homepage - warwick.ac.uk/fac/sci/maths/research/events/seminars/areas/geomtop/

The talks start five minutes after the hour. Talks are typically 55 minutes long, including time for questions.

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