On the classification of Wild Proper Arcs and Knots

We consider some natural classi?cation problems for proper arcs and knots and analyze their complexity using Borel reducibility, a nowa- days standard tool in descriptive set theory. After reviewing a recent result of Kulikov on the classi?cation of wild proper arcs/knots up to equivalence appeared on the Transactions of the American Mathematical Society, we consider a binary relation called "component" which is crucial to that proof. Since such relation turns out to be a pre-order, we consider the problem of classifying (wild) arcs up to the associated equivalence relation, which may be interpreted as asserting that two arcs have "same complexity". By adapt- ing Kulikov's argument to our new setup, we ?rst show that the complexity of such relation is bounded from below by the (oft-overlooked) sub-interval relation between countable linear orders. Then we prove various new results on the complexity of the latter relation with respect to Borel reducibility, showing in particular that it is at least as complicated as the isomorphism relation between linear orders. In the same fashion, we also consider other pre-orders on proper arcs/knots which have appeared in the literature and assess their descriptive set-theoretical complexity.