Not to follow you around leaving comments about this or anything, but one of the best examples of this is the Alexander module of a knot K in S^3: here we let X be the complement S^3-K and Y its universal abelian cover. This is a Z-fold cyclic cover, so the Alexander module is a module over R=Z[Z]=Z[t,t^{-1}]. Its 1st elementary ideal (the ideal generated by the mxm minors of an nxm presentation matrix) is always principal, and its generator is the Alexander polynomial of K.

If the 3-manifold is the complement of a fibered knot K in S^3, so the fiber is a punctured surface of genus g(K), then H^1(S^3-K) is exactly Z and the characteristic polynomial of that monodromy has to be the Alexander polynomial of K.

Here's another example for the sake of originality: two 3-dimensional lens spaces L(p,q1) and L(p,q2) are homotopy equivalent iff q1*q2 = \pm n^2 (mod p) for some n, so L(5,1) and L(5,2) are not homotopy equivalent. But they both have fundamental group Z/5Z and universal cover S^3, so their homotopy groups are the same.

Beaten by 21 seconds -- I guess this really is a standard example!

One approach that might be interesting and which avoids hard unknots is to represent a knot as a grid diagram and see whether it admits any series of commutation moves followed by a destabilization (these are some of the grid diagram analogues of Reidemeister moves). It's not clear what such a series of moves would be or how to figure this out efficiently, but a knot is the unknot iff you can repeat this until you get the trivial 2x2 diagram. See Dynnikov's paper "Arc-presentations of links. Monotonic simplification", arXiv:0208153.

My solution to this problem was to only import published papers using the MathSciNet plugin, so that it would store their MR numbers. Then you can export BiBTeX, use something like grep to pull out all the numbers, and pull the correct entries off of MathSciNet with a huge query (make each field "MR Number", connected by "OR", and retrieve BibTeX citations from the results page). There's a thread on the Papers support forums that's over two years old now where people have been requesting BibTeX support for things that aren't journal articles; maybe it'll happen in version 2.0.

The wiki article says that there is a knot invariant coming from a filtration on the Heegaard Floer homology of a 3-manifold, but nothing about what that filtration is or what properties the associated object has (other than categorifying the Alexander polynomial). My long answer below was an attempt to explain some of that.

Every class in H_k(X) is realizable for k <= 6 or k >= n-2, so the first possible example is H₇(X) for X a 10-manifold. Apparently the 10-dimensional Lie group SP(2) provides such a class; this is constructed in "Cycles, submanifolds, and structures on normal bundles" by Bohr, Hanke and Kotschick, arXiv:0011178.