The asymptotic number of prime alternating links
Résumé
The first precise asymptotic result in enumerative knot theory is the determination by Sundberg and Thistlethwaite (\emph{Pac.\ J.\ Math.}, 1998) of the growth rate of the number $A_n$ of prime alternating links with $n$ crossings. They found $\lambda$ and positive constants $c_1$, $c_2$ such that \[ c_1 n^{-7/2}\lambda^n \leq A_n \leq c_2 n^{-5/2}\lambda^n. \] In this extended abstract, we prove that the asymptotic behavior of $A_n$ is in fact \[ A_n\; \mathop{\sim}_{n\rightarrow\infty} \;c_3\; n^{-7/2}\lambda^n, \] where $c_3$ is a constant with an explicit expression.