The unknot and the Hopf link are (as far as I know) the only links whose complements have abelian fundamental groups. Are there more examples whose complement have amenable fundamental group?
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No. Suppose that $K$ is a non-trivial knot. Then its knot genus is at least one. Thus $X = S^3 - K$ contains a $\pi_1$-essential surface (with boundary) of genus at least one. Thus $\pi_1(X)$ contains a free group of rank at least two.
The story for links is similar.
A more subtle proof uses the geometrisation theorem. Every link is either a split link, a torus link, a satellite link, or a hyperbolic link. Torus link complements (other than of the unknot and Hopf link) fibre over a base orbifold with negative Euler characteristic - this gives the desired free group. Hyperbolic knot complements have free groups in their fundamental groups for geometric reasons. Split (satellite) links have an essential sphere (torus) in their complement - so cut and induct.
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2$\begingroup$ The claim "Every link is either a torus link, a satellite link, or a hyperbolic link." is wrong (true for knots). $\endgroup$ Commented Mar 5, 2023 at 20:04
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1$\begingroup$ Good point. I’ve added the missing case. $\endgroup$– Sam NeadCommented Mar 5, 2023 at 22:32