Knots in $S^1\times S^2$ 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?
 A: 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. 
