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# The Lehmer Polynomial and Pretzel Links

Published:2001-12-01
Printed: Dec 2001
• Eriko Hironaka
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## Abstract

In this paper we find a formula for the Alexander polynomial $\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with $(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and $p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7} (x)$ is the well-known Lehmer polynomial, which is conjectured to have the smallest Mahler measure among all monic integer polynomials. We confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.
 Keywords: Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups
 MSC Classifications: 57M05 - Fundamental group, presentations, free differential calculus 57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45} 11R04 - Algebraic numbers; rings of algebraic integers 11R27 - Units and factorization

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