Begging your pardon for indulging in some personal history (perhaps personal propaganda), I will explain how I ended up
applying R'ecollte et Semaille. I do apologize in advance for interpreting the question in such a self-centered fashion!
I didn't come anywhere near to reading the whole thing, but I did spend many hours
dipping into various portions while I was a graduate student. Serge Lang had put his copy into the
mathematics library at Yale, a very cozy place then for hiding among the shelves and getting lost
in thoughts or words. Even the bits I read of course were hard going. However, one thing was clear
even to my superficial understanding: Grothendieck, at that point, was dissatisfied with motives. Even though I wasn't knowledgeable enough to have an opinion about the social commentary in the book, I did wonder quite
a bit if some of the discontent could have a purely mathematical source.
A clue came shortly afterwards, when I heard from
Faltings Grothendieck's ideas on anabelian geometry. I still recall my initial reaction to the section conjecture: `Surely there
are more splittings than points!' to which Faltings replied with a characteristically brief question:' Why?'
Now I don't remember if it's in R&S as well, but I did read somewhere or hear from someone that
Grothendieck had been somewhat pleased that the proof of the Mordell conjecture came from outside
of the French school. Again, I have no opinion about the social aspect of such a sentiment (assuming the story true), but it is
interesting to speculate on the mathematical context.
There were in Orsay and Paris some tremendously powerful
people in arithmetic geometry.
Szpiro, meanwhile, had a very lively interest in the Mordell conjecture, as you can see from his
writings and seminars in the late 70's and early 80's. But somehow, the whole thing didn't come together.
One suspects that the habits of the Grothendieck school,
whereby the six operations had to be established first in every situation
where a problem seemed worth solving, could be enormously helpful in some situations, and limiting in some others. In fact, my impression is that Grothendieck's discussion of the operations in R&S has an ironical tinge. [This could well be a misunderstanding due to faulty French or faulty memory.]
Years later, I had an informative conversation with Jim McClure at Purdue on the demise of sheaf theory in topology. [The situation has
changed since then.] But already in the 80's, I did come to realize that the motivic machinery didn't fit in very well
with homotopy theory.
To summarize, I'm suggesting that the mathematical
content of Grothendieck's strong objection to motives was inextricably linked with his ideas on homotopy theory as appeared in 'Pursuing Stacks' and the anabelian letter to Faltings, and catalyzed by his realization that the motivic philosophy had been of limited use (maybe even a bit of
an obstruction)
in the proof of the Mordell conjecture. More precisely, motives were inadequate for the study of points (the most basic maps between schemes!) in any non-abelian setting, but Faltings' pragmatic approach using all kinds of Archimedean techniques may not have been quite Grothendieck's style either. Hence, arithmetic homotopy theory.
Correct or not, this overall impression was what I came away with
from the reading of R&S and my conversations with Faltings, and it became quite natural to start thinking about a workable approach to
Diophantine geometry that used homotopy groups. Since I'm rather afraid of extremes, it was pleasant to find out eventually that
one had to go back and find some middle ground between the anabelian and motivic philosophies to get definite results.
This is perhaps mostly a story about inspiration and inference, but I can't help feeling like I did apply R&S in some small way. (For a bit of an update, see my paper with Coates here.)
Added, 14 December: I've thought about this question on and off since posting, and now I'm quite curious about the bit of R&S I was referring to, but I no longer have access to the book. So I wonder if someone knowledgeable could be troubled to give a brief summary of what it is Grothendieck really says there about the six operations. I do remember there was a lot, and this is a question of mathematical interest.
Best Answer
It wasn't the properties of knots, but rather the hydrodynamical properties of closed fluids. It stems from the most basic facts in fluid mechanics:
Kelvin proved (assuming inviscid flows) that a closed curve $C$ of fluid particles (velocity field $u$) has its circulation $\oint_Cu\cdot dl$ independent of time. His theorem isn't true if the curve is fixed in space -- it literally must be a material curve that can move with the fluid.
This is closely related to Helmholtz's laws, which say that vortex tubes are frozen into the fluid. The relation: vorticity is by definition $\omega=\nabla\times u$, and Stokes' theorem shows that the flux of vorticity $\int_S\omega\cdot dS$ is precisely the circulation along slices $S$ of the vortex tube.
As a corollary, interlinked vortex tubes preserve their topology when being pushed around. It was this "state of permanence" that led to Kelvin's weird theory.