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

Question

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.

Answer 1

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 $G$.

Answer 2

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.