Link Invariants
Examples
Related concepts
transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
Arithmetic topology is a theory describing some surprising analogies between 3-dimensional topology and number theory (arithmetic), where knots embedded in a 3-manifold behave like prime ideals in a ring of algebraic integers. See also at Spec(Z) – As a 3d space containing knots.
Under this analogy, the 3-sphere, $S^3$ corresponds to the ring of rational numbers $\mathbb{Q}$, or rather (the closure of) $spec(\mathcal{O}_{\mathbb{Q}})$ (i.e., $spec(\mathbb{Z})$), since the 3-sphere has no non-trivial (unbranched) covers while $\mathbb{Q}$ has no non-trivial unramified extensions. The linking number between two embedded knots in the 3-sphere then corresponds to the Legendre symbol between two primes in the ordinary integers.
The so-called M^2KR dictionary (Mazur-Morishita-Kapranov-Reznikov) relates terms from each side of the analogy (see sec 2.2 of Sikora).
Note: Regarding (4), some have argued that $Cl(K)$ should correspond to the full first integral homology group, (see, e.g., Goundaroulis & Kontogeorgis).
The correspondence between $\pi^{et}_1(\mathbb{Z} -\{p\})$ and $\pi_1(S^3 \setminus K)$ can be developed to relate the Legendre symbol for two primes to the linking number of two knots, and further to the Rédei symbol for three primes and Milner’s triple linking number. Thus we can find a ‘Borromean link’ of primes, such as $(13, 61, 397)$, where each pair is unlinked.
Similar to M^2KR, but with the introduction of a 2-dimensional foliation on the 3-manifold and a flow such that finite primes $p$ correspond to periodic orbits of length $log N p$ and the infinite primes correspond to the fixed points of the flow (Deninger02). (See also the work of Baptiste Morin on the Weil-étale topos.)
Reznikov has modified the dictionary (Reznikov 00, section 12) so as to associate a number field with what he calls a $3\frac{1}{2}$-manifold, that is a closed three-manifold $M$, bounding a four-manifold $N$, such that the map of fundamental groups $\pi_1(M) \to \pi_1(N)$ is surjective.
Barry Mazur observed that for an affine spectrum $X = Spec(D)$ of the ring of integers $D$ in a number field, the groups $H^n_{et}(X, \mathbb{G}_{m, X})$ vanish (up to 2-torsion) for $n \gt 3$, and is equal to $\mathbb{Q}/\mathbb{Z}$ for $n = 3$, where $\mathbb{G}_{m, X}$ is the étale sheaf on $X$ defined by associating to a connected finite étale covering $Spec(B) \to X$ the multiplicative group $\mathbb{G}_{m, X}(Y) = B^{\times}$.
Also, there is a non-degenerate pairing for any constructible abelian sheaf $M$,
where $M^{'} = Hom(M, \mathbb{G}_{m, X})$. This resembles Poincaré duality for 3-manifolds.
Minhyong Kim argues that the normal bundle of an embedding of a circle corresponding to a prime in $Spec(\mathbb{Z})$ is 2-dimensional (Kim).
Baptiste Morin claims to provide a unified treatment via equivariant etale cohomology (Morin06).
The virtually fibered conjecture says that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface fiber bundle over the circle.
Christopher Deninger, A note on arithmetic topology and dynamical systems, (arxiv:0204274)
Dimoklis Goundaroulis, Aristides Kontogeorgis, On the Principal Ideal Theorem in Arithmetic Topology, (talk, paper)
Minhyong Kim, note
Thomas Koberda, Class Field Theory and the MKR Dictionary for Knots, (pdf)
Baptiste Morin, Applications of an Equivariant Etale Cohomology to Arithmetic Topology, arxiv:0602064
Masanori Morishita, Analogies between Knots and Primes, 3-Manifolds and Number Rings, (arxiv:0904.3399)
Masanori Morishita 2012, Knots and Primes: An Introduction to Arithmetic Topology, Springer
Alexander Reznikov, Embedded incompressible surfaces and homology of ramified coverings of three-manifolds, Selecta Math. 6(2000), 1–39
Adam Sikora, Analogies between group actions on 3-manifolds and number fields, (arxiv)
Masanori Morishita, Knots and Primes: An Introduction to Arithmetic Topology, 2012 (web)
Christopher Deninger, A note on arithmetic topology and dynamical systems (arXiv:math/0204274)