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This question is somewhat a continuation of the question Gluing a solid torus to a solid torus with annulus inside.

If we consider a genus one handlebody $U$ with an (nontwisted)annulus inside the handlebody so that its first homology is not zero in the handlebody.

Let $V$ be another genius one handlebody. Let $f$ be a Dehn twist about a meridian of the boundary of $U$. We think this map $f$ as a homeomorphism from the boundary of $U$ to the boundary of $V$.

Let $U \cup_f V$ be a result of gluing via $f$, which is isotopic to $S^1\times S^2$. Then the annulus is in $U \cup_f V$ somehow.

We obtain the same manifold as follows. First, we do a twist on $U$ along a meridian (extending the Dehn twist) and then glue it with $V$ via an identity map of the boundaries.

Then in this case if we look at the annulus in the resulting manifold, it look like locally (in the $U$) a twisted annulus since we did twist first.

But in the first case, the annulus is not twisted locally.

So my questions are;

  1. Is there a canonical way to define a twist or framing in a 3-manifold?
  2. What is a good way to look at knots or annulus in a 3-manifold? (as the above example, we need some method to look at knots in a 3-manifold.)

Any help or references are appreciated. Thank you in advance.

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up vote 4 down vote accepted

What does it mean to glue two manifolds $M_{1,2}$ along a "common" boundary $V$? Well, what do we mean by "common boundary"? We mean a pair of homeomorphisms $\varphi_{1,2}\colon\, V\to \partial M_{1,2}$, and the gluing map $f$ is a self-homeomorphism of $V$. This is how gluing is done formally- twisting $U$ doesn't make any sense, because $U$ isn't a submanifold of anything.

The result of the gluing is a manifold $X$ with two embeddings $\psi_{1,2}\colon\, M_{1,2}\to X$ such that $X=M_1\cup M_2$ and $M_1\cap M_2=\psi_{1}(\partial M_{1})=\psi_1(\varphi_1\circ f\circ\varphi_2^{-1}(\partial M_2))=\psi_2(\partial M_2)$.

So the answer is no- a manifold is not an embedded object, so it doesn't make sense to twist it or to knot it.

For how to define framings in a 3-manifold, see here, and the linked discussion in Gompf-Stipsicz.

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