Knots and Primes

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Knots and Primes OSAZ SS 14 18. März 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 between both areas could be bridged. We will start to recollect some basic notions and results from algebraic topology and number theory in the first three talks. The fourth talk will present fundamental analogies between 3-manifolds and number rings, knots and primes 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.

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