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I'm looking at the following topology qualifying exam question:

Let K be a knot in $S^3$. Construct a map $f: S^2 \vee S^1 \rightarrow (\mathbb{R}^3 - K)$ that induces an isomorphism of integral homology.

My biggest problem is I'm not really sure what a knot in $S^3$ is. Is it just an embedding of a circle into $S^3$? Is there a nice reference for these? Also, a hint as to what the map should be would be appreciated. Should I define it on a cellular level?

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Indeed, a knot is just a copy of $S^1$ in $S^3$. Most frequently, one considers only tame knots, which we can define variously to be the polygonal or smooth ones (of course, the two options are not equivalent, but they capture more or less the same class of examples), excluding the so called wild knots.

You should try to get a hint yourself :)

Try to construct such a map when $K$ is a standardly embedded circle (the so called unknot), then try a simple knot like the trefoil (google will find a picture for you), and by then you will know how to deal with the general case.

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    No it does not deformation retract in general. The homotopy group of the knot complement is not abelian in general, the the fundamental group of why you are tryin to deform it into is $\mathbb Z$.2011-12-23