The asymptotic number of prime alternating links

Colloque avec actes et comité de lecture. internationale.International audienceThe 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