# How to visulize surface link in four dimension?

I am now facing a problem with "surface link" in four dimension. I have heard that three 2-torus can be linked in four dimension. And I have created a movie by cutting four dimensional space with series of parallel three dimension spaces. However, I still do not know how to visualize the link. In other words, which phenomena in the movie can tell me these three 2-torus are really linked?

Besides, is there general "knot theory of surface" in four dimension space? Is there topological invariant (such as Jones polynomial) to characterize "knot" in four dimension?

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Do you mean a single $S^1 \times S^1$ in $\mathbb R^4$? People wouldn't use the terminology linked for that, generally they'd refer to that as knotting. If there's more than one $S^1 \times S^1$, then people refer to linking. – Ryan Budney May 3 '13 at 5:03
There's many invariants available, for example there is an Alexander polynomial. – Ryan Budney May 3 '13 at 5:03

2) Yes, there is a knot theory of surfaces in $\mathbb R^4$. Perhaps start by reading standard references, like Hillman's book on knot theory?
3) There is an Alexander polynomial. But no invariant is known to fully characterize knots -- the fundamental group of the complement is quite strong. The 2nd homotopy group as a module over $\pi_1$ is also fairly useful but sometimes difficult to compute. There are duality pairings and such.
Regarding your comment on the Jones polynomial -- the Jones polynomial for links in $\mathbb R^3$ is not known to characterize knots in any way. It's a fairly strong invariant in terms of knots and links in low-crossing censuses, but many things share the same Jones polynomial.