Here’s something I didn’t know. Suppose we consider the space K of “long knots” — embeddings of R into R^3, which send t to (t,0,0) whenever |t| > 1. By closing up the large |t| ends of the arc, you get a knot in S^3.
The path component of K containing the embedding of R along the x-axis is the space of long unknots. Hatcher proved, as a consequence of his proof of Smale’s conjecture, that the space of long unknots is contractible! Hatcher also proved that the space of short unknots is homotopic to the Grassmannian of 2-planes in 4-space. (I take it this follows from the contractibility of the space of long unknots, but didn’t think about it.) Moreover, Hatcher proves in the unpublished “Topological Moduli Spaces of Knots” that every connected component of K is a . When the component corresponds to a torus knot, the fundamental group is Z: for a hyperbolic knot, it is ZxZ.
How do we get a circle in the space of long knots? Hatcher makes a lovely “moving bead” argument on p.3 of the linked preprint. Let S be a “short knot” of the given type, i.e. an embedding of S^1 into S^3. For each point x on S, draw a small bead B around x; then the complement of B in S^3 looks like R^3, and the segment of the knot outside the bead is a long knot. Now let the bead slide around the knot. For each x, you get a long knot in the same isotopy class, and this gives a circle contained in the given connected component of K.
I was thinking about this because of Greg Buck‘s very interesting colloquium yesterday. Greg is interested in the space of knots with positive thickness — what you might call the space of knotted ropes. Let’s just think about the unknot. For any knot K in R^3, we can define the radius of K to be the minimal distance between a point x on K and another point on K which lies on the plane through x perpendicular to the knot. Let U_r be the space of unknots of radius r. U_0 is just the space considered by Hatcher above, a Gr(2,4). As we increase r, the space U_r gets smaller — at some point, it vanishes entirely. Buck presented a physical argument that U_r is disconnected for some intermediate values of r: that is, he passed around a thick closed rope which couldn’t be untangled, but whose meridian is an unknot.
Beyond that, how does the topology of U_r vary with r? I have no idea — but what a beautiful question! It is, of course, very reminiscent of questions about configurations of hard discs in a box discussed here earlier.
Buck’s talk centered on an energy functional on the space of knots, which blows up when the knot gets very close to acquiring a self-intersection (i.e. when the radius gets small.) You might think of as measuring “distance from the boundary of moduli space.” You might even hope that would be some kind of Morse function on the components of the space of knots! Indeed, Hatcher expresses a desire for exactly such a function, which would provide a retract of the infinite-dimensional space of knots of a given isotopy type to some finite-dimensional submanifold of minimal energy whose topology we could hope to understand. At least in some simple cases, Buck’s energy seems to behave like such a function; he showed us magnificent movies of a crimped, tangled knot flowing along the energy gradient to a handsome, easy to grasp, maximal-radius representative of its isotopy class. Great!