# The construction of knotted surfaces in $\mathbb{R}^4$

For a two-sphere embedded in $\mathbb{R}^4$,how can you check whether or not there is an ambient isotopy to the "standard" 2-sphere (the set of points $(x,y,z,0)$ in $\mathbb{R}^4$ distance 1 from the origin)?

Knot theory was discussed in an intro topology course I took and I'm wondering about further generalizations of the concept. It seems to me that such "knotted" 2-spheres could be created by rotating a knotted arc about a plane in $\mathbb{R}^4$; and knotted tori by rotating a standard knot about a nonintersecting plane. However I cannot think of a way to show which of these constructions, if any, are indeed nonisotopic to their standard counterparts.

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The Alexander polynomial is one of the more interesting isotopy-invariants for co-dimension two embedded things in spheres. So it applies to both of the things you're interested in -- knotted $2$-spheres and tori in $S^4$. Take a look at Akio Kawauchi's Survey of Knot Theory, or Jonathan Hillman's Algebraic Invariants of Knots and Links for some details. The Alexander polynomials have more subtle, associated invariants like Tristram-Levine invariants and such, these are induced from Poincare duality.
The "rotating about a plane" construction you mention is called Artin spinning. It produces non-trivial knots for fairly simple reasons. Given a knot $K$ in $\mathbb S^3$, let $G$ be the fundamental group of its complement. Then if you do Artin spinning on $K$ to produce a knotted $S^2$ in $S^4$, call it $K'$, then the fundamental group of the complement of $K'$ is also isomorphic to $G$.