I know that $H_1$ of the complement of a knot or a link can be obtained by taking the commutative quotient group which can be computed by Wirtinger presentation theorem. My questions are following
- I have shown that for a knot, the first homology group is the infinitely cyclic group, is this also true for a link?
- When I consider the higher homology groups, I tried to calculate them by M-V sequence, but it seems not work, so I wonder if there are some ways to compute them.
Thanks advanced!
Alexander Duality is you friend here. It relates the homology of the complement with the cohomology of the link itself, which is just a disjoint union of circles. In particular, $H_1$ has one $\mathbb Z$ for every link component. $H_2$ has one fewer copy of $\mathbb Z$, essentially because you need to use reduced homology in the statement of the Alexander Duality isomorphism.
MV should provide an alternate proof of this, if you take $\mathbb R^3=U\cup V$ with $U$ being a tubular neighborhood of the link, and $V$ being the link complement.