  We investigate a class of combinatory algebras, called ribbon combinatory
algebras, in which we can interpret both the braided untyped linear lambda
calculus and framed oriented tangles. Any reflexive object in a ribbon category
gives rise to a ribbon combinatory algebra. Conversely, From a ribbon
combinatory algebra, we can construct a ribbon category with a reflexive
object, from which the combinatory algebra can be recovered. To show this, and
also to give the equational characterisation of ribbon combinatory algebras, we
make use of the internal PRO construction developed in Hasegawa's recent work.
Interestingly, we can characterise ribbon combinatory algebras in two different
ways: as balanced combinatory algebras with a trace combinator, and as balanced
combinatory algebras with duality.
