In the 1960s, Barry Mazur noticed an analogy between knots and primes that led to a view of the main conjecture of Iwasawa theory in which the p-adic L-function plays the role of the Alexander polynomial. In the late 1980s, Edward Witten reconstructed the Jones polynomial as a path integral-in the sense of quantum field theory-associated to SU(2) Chern-Simons theory. This talk will review some of this history and present a weak arithmetic analogue of Witten’s formula. (This is joint work with Magnus Carlson, Hee-joong Chung, Dohyeong Kim, Jeehoon Park, and Hwajong Yoo.)