This is essentially what I suspect, also. The curve C, as is easy to see, has winding number +1 about 2 of the holes, and 0 about the other one. (Didn't Allan mention this, though?) The curve C can be obtained by starting with a simple closed curve C_0 enclosing Q and R but not P. Then some homeomorphism h: R^2 - {P,Q,R) -> R^2 - {P,Q,R} (determined up to isotopy) takes C_0 to C. The group of orientation-preserving homeomorphisms of R^2 - {P,Q,R} -> R^2 - {P,Q,R}, up to isotopy, is generated by Dehn twists about {P,Q} and about {Q,R} . . . though I neglected to say "orientation-preserving" earlier. These twists if chosen well will leave a basepoint * fixed, and thus the group Isot(R^2 - {P,Q,R}) will define a homomorphism phi: Isot(R^2 - {P,Q,R}) -> Aut(Pi_1(R^2 - {P,Q,R}, *) via phi([h])([L]) = h_#([L]), (h_# denoting h on the level of pi_1 for any homeomorphism h: R^2 - {P,Q,R} -> R^2 - {P,Q,R} that is the composite of a finite sequence of these Dehn twists, and any loop L based at *). If C_0 contains *, then so will C, which can then be expressed as an element of pi_1(R^2 - {P,Q,R}, *), i.e., the free group on the loops a, b, c based at *, going around P, Q, R -- presumably the word accompanying the painting. (Which certainly was a lot of words to say very little about one.) --Dan Rich wrote: << I'll hazard a guess that the curve in the picture is supposed to be the homotopic sum of loops around the individual punctures, and that the Word is the formula for the curve. Perhaps A is a simple counterclockwise loop around the first puncture, etc. If so, the transcription has a couple of problems: For the winding number of the curve to be 0 around each puncture, the total exponent of A should be 0 (it is), and ditto for B and C. (Nope: There are extra B' and C'.) IIRC, a contractable loop would simplify to Identity when A,B,C are interpreted in the (non-commutative) free group on three generators. I'm uneasy about this interpretation, since the rules for homotopy of curves allow the curve to cross itself, while the puzzle at hand forbids this.

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