Every classical or virtual knot is equivalent to the unknot via a
sequence of extended Reidemeister moves and the so-called forbidden moves. The
minimum number of forbidden moves necessary to unknot a given knot is a new
invariant we call the forbidden number. We relate the forbidden number to several
known invariants, and calculate bounds for some classes of virtual knots. This is
joint work with Sandy Ganzell and Blake Mellor.
We give a classification of open Klein topological conformal field theories in terms of Calabi-Yau A_\infty-categories endowed with an involution. Given an open Klein topological conformal field theory, there is a universal open-closed extension whose closed part is the involutive variant of the Hochschild chains of the open part.