Arcs and knots via linear and circular orders

Knots are very familiar and tangible objects in everyday life and play an important role in modern mathematics. A mathematical knot is a homeomorphic copy of S_1 embedded in S_3. Proper arcs are intuitively obtained by cutting a knot and are defined as copies of the unit interval embedded in a closed ball. Following an earlier paper by Weinstein (then called Kulikov), we use discrete objects, such as linear and circular orders, to gain insights into arcs and knots. To this end we study in detail the relation of convex embeddability between countable linear and circular orders. This leads to results about the combinatorial and descriptive set theoretic complexity of natural subarc and subknot relations. We point out that knot theory usually considers only tame knots, while we are dealing mainly with wild knots. Joint work with Martina Iannella, Luca Motto Ros and Vadim Weinstein