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.

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