# Questions tagged [arithmetic-topology]

Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds.

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### Is Mazur's analogy between arithmetic and topology formal, in any sense?

I preface my question by admitting I know no algebraic geometry nor algebraic number theory. I do know some algebraic topology. I'm a student.
Recently I learned about sheaf cohomology. Then a little ...

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### Lengths of tangles and prime numbers

Akshay Venkatesh recently gave a nice public lecture at the IAS on the analogy between primes and knots.
The analogy involves a conversion factor, which is described in the talk as follows: given a ...

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### Why does the longitude correspond to Frobenius in Arithmetic Topology, and other strange phenomena

I am trying to adress Morishita's book Knots and Primes to discover a bit about Arithmetic Topology, but some analogies puzzle me. I know that the correspondence should be addressed with a grain of ...

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### Is there an "arithmetic cobordism category"?

This question is a clumsy attempt to apply a certain analogy. I hope that if the answer is negative it comes with a clarification of the scope and limitations of the analogy.
Arithmetic topology is ...

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### Zeta function for curves in a manifold

Motivation
In the analogy between prime numbers and knots, the prime number is thought sometimes as the circle of length $l([p]) = \text{log}\,p$. This is so you can express the zeta function as
$$ \...

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### Spec Z analogue of Thurston program?

It's been known for a while that primes in number fields can be thought of, from an algebraic point of view, to be similar to knots in 3-manifolds. A good reference (thanks to this question) would be ...

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### Questions about analogy between Spec Z and 3-manifolds

I'm not sure if the questions make sense:
Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...