The aymptotic number of prime alternating links

Sébastien Kunz-Jacques, Gilles Schaeffer

To appear at Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe, Arizona (USA), May 20-26, 2001


The first precise asymptotic result in enumerative knot theory is the determination by Sundberg and Thistlethwaite (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.

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