One can define the linking number of disjointly embedded curves $K,L\subset S^{3}$ in a variety of ways, as is discussed in Chapter 5.D of Rolfsen's "Knots and Links". One way is the Gauss Integral
$$\mathrm{lk}(K,L) = \frac{1}{4\pi}\int_{K\times L}\dfrac{\mathbf{x}-\mathbf{y}}{|\mathbf{x}-\mathbf{y}|^3}\cdot\mathrm{d}\mathbf{x}\times \mathrm{d}\mathbf{y}$$
(or symbols to that effect). This will be an integer when the curves are closed, and a real number in general.
The integral formula has been generalised to deal with the case of disjointly embedded closed manifolds $K^{k} ,L^{\ell}\subset S^{k+\ell+1}$ (see here and here for example). Presumably these formulas output real numbers when the manifolds $K$ and $L$ have boundaries.
My question concerns the situation of disjointly embedded submanifolds $K^k,L^\ell\subset S^n$, where $k+\ell >n-1$. Is there a useful notion of linking number in this case? For instance, take a surface and a curve in $3$ dimensions (so $k=2$, $\ell=1$ and $n=3$ in the above). Then we could try to define the linking number by somehow "integrating" $\mathrm{lk}(\gamma,L)$ over closed curves in $\gamma\subset K$.
This generalised linking number should be able to measure, say, how many times a curve passes through a length of tube.
Have such things been considered useful before? Or am I just talking nonsense?
Added: As Ryan points out in his comment, I'm not really looking for an isotopy invariant. Also (thanks to Kevin and Tom's answers) I'm slowly coming round to the idea that a single number won't really tell you much about the relative positions of the manifolds, but maybe a matrix valued function (with rows and columns indexed by homology bases for $K$ and $L$ in the appropriate dimensions) might be useful.
Best Answer
Sort of obvious, but: in general you get lots of linking numbers. If spaces $K$ and $L$ are mapped disjointly into $\mathbb R^n$ then for every $a\in H_i(K)$ and $b\in H_j(L)$ with $i+j=n-1$ you get a number. It can be obtained by pulling back a generator of $H^{n-1}(S^{n-1})$ via the evident map $K\times L\to S^{n-1}$ and evaluating the resulting element of $H^{n-1}(K\times L)$ on $a\times b\in H_{n-1}(K\times L)$. (If $a$ and $b$ are represented by smooth manifolds then this can be expressed as an integral.) This is a bilinear pairing $H_i(K)\times H_{n-i-1}(L)\to \mathbb Z$.