Primes and Knots.

Intro -- Contents -- Preface -- Categorification of the skein module of tangles -- The double shuffle relations for p-adic multiple zeta values -- Galois p-groups unramified at p - A survey -- On capitulation theorems for infinite groups -- Multiple zeta values and Grothendieck-Teichmüller groups -- Asymptotics of q-difference equations -- 1. Introduction -- 1.1. The goal -- 1.2. The colored Jones function -- 1.3. The Hyperbolic Volume Conjecture -- 1.4. q-difference equations -- 1.5. Asymptotics of differential equations with a parameter -- 1.6. Asymptotics of difference equations -- 1.7. Asymptotics of difference equations with a parameter -- 1.8. Statement of the results -- 1.9. What's next? -- 1.10. Acknowledgement -- 2. є-difference equations -- 2.1. є-difference equations -- 2.2. Converting q-difference equations to є-difference equations -- 3. Some linear algebra -- 4. Existence of formal solutions -- 4.1. An alternative formal series -- 5. Proof of Theorem 3 -- 5.1. Existence of a solution corresponding to the eigenvalue of maximum magnitude -- 5.2. A reduction to an є-difference equation of smaller degree -- 5.3. The solutions form a locally fundamental set -- 6. Regular solutions and their asymptotics -- 6.1. Regular solutions to є-difference equations -- 6.2. Asymptotics of regular solutions of є-difference equations -- 6.3. Asymptotics of regular solutions of q-difference equations -- 7. Applications to Quantum Topology -- 7.1. The A-polynomial of a knot and its noncommutative version -- 7.2. Examples: The 31 and 41 knots -- References -- The mapping class group acts reducibly on SU(n)-character varieties -- Pro-p link groups and p-homology groups -- Introduction -- 1. Pro-p completion of a link group -- 2. p-adic Milnor invariants -- 3. Completed Alexander modules.

4. Galois module structure of the p-homology group of a p-fold cyclic branched cover -- 5. Iwasawa type formulas for the p-homology groups of pm-fold cyclic branched covers -- A quantum introduction to knot theory -- Classical knot invariants and elementary number theory -- Harmonic and equianharmonic equations in the Grothendieck-Teichmüller group, II -- On p-adic properties of the Witten-Reshetikhin-Turaev invariant -- Seiberg-Witten integrable systems and periods of rational elliptic surfaces -- On the finiteness of various Galois representations -- Some new-type equations in the Grothendieck-Teichmüller group arising from geometry of Mo,5.

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