Abstract

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 lim_n→∞(A_n) is obtained, where A_n is the number of n-crossing, prime, unoriented, alternating link types. This result follows from a detailed investigation of the sequence (a_n), where a_n 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 ∑ a_nzⁿ is shown to satisfy a certain functional equation; a study of the analytic properties of this equation yields an asymptotic formula for a_n, and a study of its algebraic properties yields a practical method for calculating a_n exactly up to several hundred crossings.

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