We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten’s Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related with invariants of generic plane curves defined by Arnold recently with deep m...

We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves, and it is related with the invariants of generic plane curves recently defined by Arnold, with ...

i.e., the supremum of all orders of torsion points on elliptic curves defined over some degree d number field. Write T (d)′ for the set of prime divisors of elements of Td, and P (d) for the largest element of T (d)′. Let TCM(d) (resp. TIM(d)) be the subset of T (d) corresponding to elliptic curves with complex multiplication (resp. with algebraic integral modulus j(E)), and similarly adding th...

The Coleman integral is a p-adic line integral. Double Coleman integrals on elliptic curves appear in Kim’s nonabelian Chabauty method, the first numerical examples of which were given by the author, Kedlaya, and Kim [3]. This paper describes the algorithms used to produce those examples, as well as techniques to compute higher iterated integrals on hyperelliptic curves, building on previous jo...