  Families of alternating knots (links) and tangles are studied using as
building block the conway defined as the twisting of two strands. The regular
representation of knots assumes the projection has the minimal number of
overpassings, and the minimal number of conways. The continued fraction
associated to rational knots is represented by gaussian brackets and products
of 2-dimensional matrices. This gives birth to an algebra of rational knots and
tangles which is easily generalized to alternating knots. A collection of 65
families of prime alternating knots with one to six conways is found. Eleven
families with six conways show peculiar behavior not present in families with a
lower or equal number of conways.
