Knot theory on arbitrary manifolds.

Question

I have been interested in knot theory recently but am still struggling with it. In the meantime I was wondering if it is sensible to talk about knots on arbitrary manifolds like say lie groups. Can I classify the knots appearing in such lie groups?

Answer 1

Yes, it is sensible to talk about knots in an arbitrary smooth manifold (or, say, PL manifold). These would be smooth embeddings $S^1\to M$ for a manifold $M$ up to ambient isotopy. You would usually want for $M$ to be at most $3$-dimensional, since for higher dimensions the classification problem reduces to homotopy theory.

For example, consider $M=G=SL_2(\mathbb{R})$. One can understand the topology of this group by considering the action of $G$ on the punctured plane $P=\mathbb{R}^2-\{(0,0)\}$, defined by matrix-vector multiplication. The vector $(1,0)$ is stabilized by matrices of the form $\begin{pmatrix}1&*\\0&1\end{pmatrix}$, and the stabilizers of other matrices are conjugates of this, since $G$ acts transitively on $P$. Hence, $G$ can be thought of as being part of a fiber bundle $G\to P$ with fiber $\mathbb{R}$. The only orientable line bundles over $P$ are all homeomorphic to an open solid torus $S^1\times \operatorname{int}(D^2)$. Adding in the boundary does not change the knot theory, so the knot theory of $SL_2(\mathbb{R})$ is equivalent to the knot theory of $S^1\times D^2$.

One way that the theory is different for $S^1\times D^2$ is that there are non-nullhomotopic knots (and the homotopy type is a knot invariant). For example, the curve $S^1\times\{(0,0)\}$, which is not an unknot. Another is that there are self-homeomorphisms of the solid torus (Dehn twists) that change the knot type. For example, Dehn twists add twists to the pattern for the Whitehead double:

One way to classify knots in solid tori is to take a disk $\{t\}\times D^2$ that intersects the knot transversely in $n$ points, slice the torus along this disk, and then instead study $2n$-tangles along with special moves that come from the "torus closure." So, one can enumerate all such knots.

In the case of $M=G=SU(2)$, then since this is homeomorphic to $S^3$, this is classical knot theory.