All knot homology theories I've seen share a flaw: their definitions explicitly use some combinatorial choices (such as a diagram presentation). The coin, however, has two sides and the other one is potential computability.

From what I've heard about the subject, a nice homology for knots should be (at least) bigraded and functorial with respect to cobordisms. Moreover, we could expect that a homomorphism corresponding to a cobordism shifts one grading by Euler characteristic of the cobordism. Also, this grading should be somehow related to the four-ball genus.

I was thinking if we can introduce a knot homology which has a natural definition (i.e. doesn't make use of combinatorial choices) and came up with the following rather tautological proposal.

For a knot $K$ in a 3-manifold $M$ consider a space $S_c(K,M)$ of all connected surfaces with Euler characteristic $c$ in $M\times[0,1)$ whose boundary is $K\times\{0\}.$ Define $\mathcal{H}_{c,j}(K)$ as $H_{j}(S_c(K,M)),$ where $H_{*}=\bigoplus_j H_j$ denotes your favorite homology theory of topological spaces.

Clearly, $\mathcal{H}_{*,*}$ satisfies the above-mentioned properties of a nice knot homology theory. Unfortunately, it seems to be very far from being computable.

To remedy things a little bit we can slightly modify the definition by fixing an unknot $U$ and taking $S_c(K,M)$ to be the space of all connected surfaces with Euler characteristic $c$ in $M\times{[0,1]}$ whose boundary is the union of $K\times\{0\}$ and $U\times\{1\}$. This transforms $\mathcal{H}_{*,*}(U,M)$ into an algebra and $\mathcal{H}_{*,*}(K,M)$ into its module.

The main question:

Is this module finitely generated?

A concrete question:

What is the (Hopf) algebra structure of $\mathcal{H}_{*,*}(U,S^3)$?

A vague question:

Do you see some spectral sequence from a known homology to $\mathcal{H}_{*,*}$?

Apart from the choice of $H_*$ there is another variation where we would consider smooth surfaces instead of topological ones (this gives a different theory since topological and smooth four-ball genera a not always same). I've tried to use some Morse theory in this variation, to no avail.