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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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    $\begingroup$ 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. $\endgroup$
    – Ryan Budney
    Commented May 3, 2013 at 5:03
  • $\begingroup$ There's many invariants available, for example there is an Alexander polynomial. $\endgroup$
    – Ryan Budney
    Commented May 3, 2013 at 5:03

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To answer your questions,

1) No individual phenomena characterises knottedness. There won't be a simple answer to this question.

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.

Another very basic invariant is the Whitney class of the normal bundle of your surface. This is an invariant that takes finitely many values.

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