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

Bibliographie

[Bour] [Bour] Bourbaki, N., Groupes et Algèbres de Lie. Vol. 13, 1968. Google Scholar

[Boyd] [Boyd] Boyd, D. W., Speculations concerning the range of Mahler's measure. Canad. Math. Bull. (4) 24 (1981), 453–469. Google Scholar

[C-W] [C-W] Cannon, J. and Wagreich, P., Growth functions of surface groups. Math. Ann. 293 (1992), 239–257. Google Scholar

[Floy] [Floy] Floyd, W. J., Growth of planar Coxeter groups, P.V. numbers, and Salem numbers. Math. Ann. 293 (1992), 475–483. Google Scholar

[F-P] [F-P] Floyd, W. J. and Plotnick, S. P., Symmetries of planar growth functions of Coxeter groups. Invent. Math. 93 (1988), 501–543. Google Scholar

[Kaw] [Kaw] Kawauchi, A., A Survey of Knot Theory. Birkhäuser-Verlag, Basel, 1996. Google Scholar

[Kir] [Kir] Kirby, R., Problems in Low-Dimensional Topology. In: Geometric Topology (ed.W.H. Kazez), Stud. Adv. Math., Amer.Math. Soc., 1997. Google Scholar

[Leh] [Leh] Lehmer, D. H., Factorization of certain cyclotomic functions. Ann. of Math. 34 (1933), 461–469. Google Scholar

[Lev] [Lev] Levine, J., Polynomial invariants of knots in codimension 2. Ann. of Math. 2 84 (1966), 537–554. Google Scholar

[Lic] [Lic] Lickorish, W. B. R., An Introduction to Knot Theory. Springer, 1997. Google Scholar

[Reid] [Reid] Reidemeister, K., Knotentheorie. Springer, Berlin, 1932. Google Scholar

[Rolf] [Rolf] Rolfsen, D., Knots and Links. Publish or Perish, Inc., Berkeley, 1976. Google Scholar

[Seif] [Seif] Seifert, H., Über das Geschlecht von Knoten. Math. Ann. 110 (1934), 571–592. Google Scholar

[Smy] [Smy] Smyth, C. J., On the product of the conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc. 3 (1971), 169–175. Google Scholar