Knots and Primes OSAZ SS 14 18. M¨arz 2014 This semester we will work out an analogy between knot and number theory, our principal reference being the book of Morishita [Mor12]. This analogy can be traced back to Gauss and his work on quadratic residues and the notion of the linking number, which he found working on electrody- namics. From then on both areas branched out independently and it was not before the geometrization of number theory (scheme theory) that the gap bet- ween both areas could be bridged. We will start to recollect some basic notions and results from algebraic topo- logy and number theory in the first three talks. The fourth talk will present fundamental analogies between 3-manifolds and number rings, knots and pri- mes which are the basis for the remaining talks. From talk five on, further analogies will be presented and a kind of dictionary will be developed, translating between number and knot theory. 1. Algebraic topology [Mor12, 2.1] 2. Arithmetic Rings [Mor12, 2.2] 3. Class Field Theory [Mor12, 2.3] 4. Knots and Primes, Linking Numbers and Legendre Symbols [Mor12, Chap. 3-4] 5. Decomposition of Knots and Primes [Mor12, Chap. 5] 6. Homology Groups and Ideal Class Groups [Mor12, Chap. 6] 7. Link Groups and Galois Groups [Mor12, Chap. 7] 8. Milnor Invariants and Multiple Residue Symbols pt. 1 [Mor12, 8.1 - 8.2] 9. Milnor Invariants and Multiple Residue Symbols pt. 2 [Mor12, 8.3 - 8.4] 1 10. Alexander Modules and Iwasawa Modules pt. 1 [Mor12, 9.1] 11. Alexander Modules and Iwasawa Modules pt. 2 [Mor12, 9.2 - 9.3] 12. Higher Order Genus Theory pt. 1 [Mor12, 10.1 - 10.2] 13. Higher Order Genus Theory pt. 2 [Mor12, 10.3 - 10.4] 14. Asymptotic Formulas [Mor12, Chap. 11] Literatur [Mor12] Morishita, Masanori: Knots and primes. Universitext. Springer, London, 2012. An introduction to arithmetic topology. 2.