The Lehmer Polynomial and Pretzel Links

Canadian mathematical bulletin

关键词: Alexander polynomial;    pretzel knot;    Mahler measure;    Salem number;    Coxeter groups;   

DOI  :  10.4153/CMB-2001-044-x

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

In this paper we find a formula for the Alexander polynomial$Delta_{p_1,dots,p_k} (x)$ of pretzel knots and links with$(p_1,dots,p_k, ega 1)$ 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 havethe smallest Mahler measure among all monic integer polynomials. Weconfirm that $Delta_{2,3,7} (x)$ has the smallest Mahler measure amongthe polynomials arising as $Delta_{p_1,dots,p_k} (x)$.

【 授权许可】

Unknown