[Most of this was written before several posts that explain much of what I've described below, but for clarity I'm leaving this unedited.] Let C be a simple closed curve in R^3 - {p,q,r}, where #{p,q,r} = 3. If the winding numbers wind(C;p) = wind(C;q) = wind(C;r) = 0, then, as Bill points out, all of p, q, r must be external to C. The interior of any simple closed curve in R^2, union the curve itself, must be a closed 2-disk. Hence C continuously deforms to a point in R^2 - {p,q,r}. This means that the situation that Jim asks about cannot happen.

From this I infer that the long complicated word written on the wall next to the simple closed curve in the complement of 3 points in R^2, in the painting by "D.S. and B.T.", is *not* a word in the fundamental group of R^2 - {p,q,r} (which is the free group on 3 generators).

Despite being 15 feet away when that painting was made, I don't recall anything I may have known about the math behind it. But I'd guess that the word is expressed in the two customary generators of the "isotopy group" G = Isot(R^2 - {p,q,r}) of R^2 - {p,q,r}. (G is the group of all self-homeomorphisms of R^2 - {p,q,r}, under composition, after identifying any two homeomorphisms that can be continuously deformed, one into the other, through homeomorphisms.) In case anyone is interested: Suppose p, q, r lie on the x-axis. Let A be a round annulus with p and q inside its inner circle, and r outside its outer circle; let B be an annulus with q and r inside its inner circle, and p outside its outer circle. Then the Dehn twist in A is the homeomorphism h_A: R^2 - {p,q,r} -> R^2 - {p,q,r} defined via h_A(x) = x for x not in A, but the circles between the inner and outer boundary components of A are each taken to themselves by a rotation by an angle that increases, with radius, from 0 to 2pi. Likewise for the Dehn twist in B, h_B: R^2 - {p,q,r} -> R^2 - {p,q,r}. Then the isotopy classes [h_A] and [h_B] are known to generate* the group Isot(R^2 - {p,q,r}). In the case of the simple closed curve of the painting, it's fun to let the exterior "ooze into" the curve, by shading that in. What remains is a very skinny and folded snake with one eye at each end (namely, the two punctures in its interior). Once this becomes clear, it's easy to imagine how a sequence of Dehn twists might result in unfolding the snake. The inverse of the corresponding word is, I'm guessing, what was painted on the wall. (But the resolution of the photo is too low for me to be able to read that word.) --Dan ___________________________________________________________________ * In general the isotopy classes of n-1 Dehn twists generate the group Isot(R^2 - {p_1,...,p_n}). Much as the n-1 transpositions (12), (23),..., (n-1 n) generate the symmetric group S_n. In fact there is an evident surjective homomorphism Isot(R^2 - {p_1,...,p_n}) -> S_n, according to the permutation of the holes {p_j}. ----------------------------------------------------------------------------- Bill T. wrote: << A simple curve in the plane is the boundary of a disk. If the disk has no points in its interior, the curve is contractible. The curve has winding number 1 or -1 about any point in the interior of the disk. The point is that in the plane with 3 or more punctures, there can be very complicated simple curves, even though they automatically have small winding numbers (0, 1 or -1) about all the punctures. [Jim P. wrote:] << What's the simplest example of a simple closed curve in the triply-punctured plane that has winding number 0 around each of the punctures but is not contractible? (Back when I was in grad school in Berkeley in the '80s, there was a painting of one such curve on the wall, along with the associated word in the fundamental group of the surface.) Also, what's the simplest way to prove that no analogous curve exists for the doubly-punctured plane?

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