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 links. How would that be with Spec Z? Then, topologists have things like virtual 3-manifolds, has that analogies in arithmetics?

An other such question:
Minhyong Kim <a href="http://londonnumbertheory.wordpress.com/2009/11/04/optimal-proofs/" title="London NT seminar blog">stresses</a>
the special complexity of number theory: "To our present day understanding, number fields display exactly the kind of order ‘at the edge of chaos’ that arithmeticians find so tantalizing, and which might have repulsed Grothendieck." Probably a feeling of such a special complexity makes one initially interested in NT. Knot theory is an other case inducing a similar impression. Could both cases be connected by the analogy above? How could a precise description of such special complexity look like and would it cover both cases? Taking that analogy, I'm inclined to answer <a href="http://londonnumbertheory.wordpress.com/2009/12/18/within-the-mess/" title="London NT seminar blog">Minhyong's question</a>
 with the contrast between low-dimensional (= messy) and high-dimensional (= harmonized) geometry. Then I wonder, if "harmonizing by increasing dimensions"-analogies in number theory or the Langlands program exist.    

Minhyong hints in a mail to "the study of moduli spaces of bundles over rings of integers and over three manifolds as possible common ground between the two situations". A google search produces an old article by Rapoport "Analogien zwischen den Modulräumen von Vektorbündeln und von Flaggen" (Analogies between moduli spaces of vector bundles and flags) (p. 24 <a href="httP://dml.math.uni-bielefeld.de/JB_DMV/JB_DMV_099_4.pdf" title="DMV Jahresbericht 99">here</a>, <a href="http://ams.mathematik.uni-bielefeld.de/mathscinet-getitem?mr=99e:14010" title="MathSciNet review">MR</a>). There, Rapoport describes the cohomology of such analogous moduli spaces, inspired by a similarity of vector bundles on Riemann surfaces and filtered isocrystals from p-adic cohomologies, "beautifull areas of mathematics connected by entirely mysterious analogies". (<a href="http://www.math.uni-bonn.de/people/orlik/Arbeiten.html" title="online book">book project</a> by R., Orlik, Dat) As interesting as that sounds, I wonder if google's hint relates to the initial theme. What do you think about it? (And has the mystery Rapoport describes now been elucidated?)

<a href="http://www.ihes.fr/~gromov/PDF/ergobrain.pdf" title="pdf">This interesting essay by Gromov</a> discusses the topic of "interestung structures" in a very general way. Acc. to him, "interesting structures" exist never in isolation, but only as "examples of structurally organized classes of structured objects", Z only because of e.g. algebraic integers as "surrounding" similar structures. That would fit to the guesses above, but not why numbers were perceived as esp. fascinating as early as greek antiquity, when the "surrounding structures" Gromov mentions were unknown. Perhaps Mochizuki has with his <a href="http://mathoverflow.net/questions/852/what-is-inter-universal-geometry" title="MO">"inter-universal geometry"</a> a kind of substitute in mind?