4 mainly TeX; added 1 characters in body


Sadly, I don't have a copy of the unpublished note, but at least I can tell you what the published paper has to do with knots.

In that paper, Mazur shows that etale cohomology for spec(ZSpec($\Z$) satisfies a sort of Poincare-duality as known for 3-dimensional manifolds. So, one might consider spec(ZSpec($\Z$) as a 3-dml 3-dimensional manifold. As its fundamental group is trivial, it should be after Poincare-Perelman the 3-sphere. Primes should be closed submanifolds in it and as the fundamental group op Z/(p) of $\F_p=\Z/p\Z$ is Zhat, $\hat\Z$, one might consider primes as circles embedded in S^3, $\S^3$, that is, as knots.

The analogy goes a lot deeper and I've written a blog-post about it some time ago (with references to Mazur's paper and follow-ups) :

Mazur's knotty dictionary

I've also given a talk about it, intended for a general public, last year. The slides are sadly in Dutch, but perhaps they convey what i I was trying to tell : (34Mb download though ...)

What does a prime number look like?

3 deleted 5 characters in body; added 1 characters in body

Sadly, I don't have a copy of the unpublished note, but at least I can tell you what the published paper has to do with knots.

In that paper, Mazur shows that etale cohomology for spec(Z) satisfies a sort of Poincare-duality as known for 3-dimensional manifolds. So, one might consider spec(Z) as a 3-dml manifold. As its fundamental group Zhatis trivial, it should be after Poincare-Perelman the 3-sphere. Primes should be closed submanifolds in it and as the fundamental group op Z/(p) is infinite cyclicZhat, one might consider primes as circles embedded in S^3, that is, as knots.

The analogy goes a lot deeper and I've written a blog-post about it some time ago (with references to Mazur's paper and follow-ups) :

Mazur's knotty dictionary

I've also given a talk about it, intended for a general public, last year. The slides are sadly in Dutch, but perhaps they convey what i was trying to tell : (34Mb download though ...)

What does a prime ideal number look like?

2 deleted 6 characters in body

Sadly, I don't have a copy of the unpublished note, but at least I can tell you what the published paper has to do with knots.

In that paper, Mazur shows that etale cohomology for spec(Z) satisfies a sort of Poincare-duality as known for 3-dimensional manifolds. So, one might consider spec(Z) as a 3-dml manifold. As its fundamental group is trivialZhat, it should be after Poincare-Perelman the 3-sphere. Primes should be closed submanifolds in it and as the fundamental group op Z/(p) is infinite cyclic, one might consider primes as circles embedded in S^3, that is, as knots.

The analogy goes a lot deeper and I've written a blog-post about it some time ago (with references to Mazur's paper and follow-ups) :

Mazur's knotty dictionary

I've also given a talk about it, intended for a general public, last year. The slides are sadly in Dutch, but perhaps they convey what i was trying to tell : (34Mb download though ...)

What does a prime ideal look like?

1