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Is there any special study of knots in this particular 3-manifold?

A more targeted / simple question: What are some nontrivial examples of knots $S^1\subset S^1\times S^2$, and is there convenient way to view them?

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    I recently saw a very cool talk which was about "knot theory from a non-knot theorist's perspective" -- it gives a framework in which to study *all* embeddings of manifolds, of which knots or links (in any given ambient manifold) are a special case. You can see my notes here: http://math.berkeley.edu/~aaron/saft/francis.pdf2012-12-04

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Typically knot theory in most small 3-manifolds reduces in various ways to knot theory in $S^3$.

For example, if the complement of a knot in $S^1 \times S^2$ isn't irreducible, there's a $2$-sphere which when you do surgery on it turns $S^1 \times S^2$ into $S^3$. So the study of these knots is simply the study of knots in $S^3$, or knots in a ball (which just happens to be in $S^1 \times S^2$).

If the complement is irreducible then the knot theory is a little different, but not all that different. For example, one way to link knot theory in $S^1 \times S^2$ to knot theory in $S^3$ is to observe that $S^1 \times S^2$ is zero surgery on the unknot. So a knot in $S^1 \times S^2$ is a Kirby diagram consisting of a 2-component link, one component is unknotted and labelled with a zero (for zero-surgery) and the other component is un-labelled. From the perspective of living inside $S^1 \times S^2$, what this amounts to doing is choosing a knot in the complement of your original knot, such that projection $S^1 \times S^2 \to S^1$ restricts to a diffeomorphism on the new knot. So there will be "Kirby moves" in addition to link isotopy needed to keep track of how knot theory in $S^1 \times S^2$ reduces to the study of these two-component links in $S^3$.

Those are two things that come to mind, anyhow.

So a standard non-trivial example in this "Kirby notation" would be the Whithead link with $0$-surgery done to one component. This gives you a non-trivial knot in $S^1 \times S^2$, the fundamental group of the complement being $\langle a, b | ab^{-2}aba^{-2}b \rangle$, which is non-abelian.

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    By the solution to the Poincare conjecture and the sphere theorem, existence of a sphere that does not bound a ball is equivalent to $\pi_2$ being non-trivial. So there's a process you can use. But more simply, any such sphere would have to lift to a sphere in a covering space -- but if you take the cover of your knot complement corresponding to $\mathbb R \times S^2 \to S^1 \times S^2$ you get a submanifold of Euclidean space $\mathbb R^3$, so you can apply the Schoenflies theorem.2012-12-04