TITLE: The construction of knotted surfaces in $\mathbb{R}^4$
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: Kawauchi's and Hillman's books are indeed good references. Many known knotted surfaces have non-trivial and easily computable fundamental groups. You should also look at R.H. Fox's "Quick Trip through Knot Theory," in Fort's Georgia Topology Conference Proceedings, republished by Dover. In addition, there are three books dedicated to studying knotted surfaces from a diagrammatic point of view. Kamada's book on Surface Braids published by the AMS, my book with Saito, "Knotted Surfaces and Their Diagrams," and my book with Kamada and Saito, "Surfaces in 4-space." 
The quandle cocycle invariants discovered by Jelsovsky, Langford, Kamada, and me are known to be powerful invariants of knotted surfaces.