When introduced to the
subject of knot theory, it is natural to ask how the number of knots and
links grows in relation to crossing number. The purpose of this article is to
address this question for the class of prime alternating links; in particular,
the exact value of limn→∞(An) is obtained, where An is the number of
n-crossing, prime, unoriented, alternating link types. This result follows from a
detailed investigation of the sequence (an), where an is the number of strong
equivalence classes of prime, alternating tangle types with n crossings (a
tangle equivalence is strong if it fixes the boundary of the ambient ball of the
tangle pointwise). The generating function ∑anzn is shown to satisfy a
certain functional equation; a study of the analytic properties of this equation
yields an asymptotic formula for an, and a study of its algebraic properties
yields a practical method for calculating an exactly up to several hundred
crossings.
is obtained, where 