Generating functions, Fibonacci numbers and rational knots

We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots. Then we show how to enumerate rational knots of given crossing number depending on genus and/or signature. This allows to determine the asymptotical average value of these invariants among rational knots. We give also an application to the enumeration of lens spaces. (c) 2006 Elsevier Inc. All rights reserved.