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The asymptotic number of prime alternating links

Sébastien Kunz-Jacques Gilles Schaeffer 1
1 ADAGE - Applying discrete algorithms to genomics
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : 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.
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https://hal.inria.fr/inria-00108015
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Submitted on : Thursday, October 19, 2006 - 3:13:03 PM
Last modification on : Friday, February 26, 2021 - 3:28:02 PM

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  • HAL Id : inria-00108015, version 1

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Sébastien Kunz-Jacques, Gilles Schaeffer. The asymptotic number of prime alternating links. Formal Power Series and Algebraic Combinatorics - FPSAC'2001, 2001, Phoenix, Arizona, 10 p. ⟨inria-00108015⟩

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