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The Lehmer Polynomial and Pretzel Links

Published online by Cambridge University Press:  20 November 2018

Eriko Hironaka
Eriko Hironaka*
Affiliation:
Department of Mathematics Florida State University Tallahassee, FL 32306 USA, email: hironaka@math.fsu.edu
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Abstract

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In this paper we find a formula for the Alexander polynomial ${{\Delta }_{p1,...,{{p}_{k}}\left( x \right)}}$ of pretzel knots and links with $\left( {{p}_{1}},...,{{p}_{k}},-1 \right)$ twists, where $k$ is odd and ${{p}_{1}},...,{{p}_{k}}$ are positive integers. The polynomial ${{\Delta }_{2,3,7}}\left( x \right)$ is the well-known Lehmer polynomial, which is conjectured to have the smallest Mahler measure among all monic integer polynomials. We confirm that ${{\Delta }_{2,3,7}}\left( x \right)$ has the smallest Mahler measure among the polynomials arising as ${{\Delta }_{p1,...,{{p}_{k}}\left( x \right)}}$.

Keywords

57M0557M2511R0411R27Alexander polynomialpretzel knotMahler measureSalem numberCoxeter groups
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 4 , 01 December 2001 , pp. 440 - 451
Copyright
Copyright © Canadian Mathematical Society 2001

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