Generating functions, Fibonacci numbers and rational knots

Details Generating functions, Fibonacci numbers and rational knots Generating functions, Fibonacci numbers and rational knots

2002-10-11

arxiv.org/abs/math/0210174v1

J. Algebra 310(2) (2007), 491--525.

Mathematics

Geometric Topology

32 pages, 3 figures

Scientific paper

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 by crossing number and genus and/or signature. This allows to determine the distribution of these invariants among rational knots. We give also an application to the enumeration of lens spaces.

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