The Lehmer Polynomial and Pretzel Links

Hironaka, Eriko

Canadian mathematical bulletin, Tome 44 (2001) no. 4, pp. 440-451

Résumé

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 ${{\Delta }_{p1,...,{{p}_{k}}\left( x \right)}}$ .

Détail

DOI : 10.4153/CMB-2001-044-x

Mots-clés : 57M05, 57M25, 11R04, 11R27, Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups