Premium
Mahler Measure of Alexander Polynomials
Author(s) -
Silver Daniel S.,
Williams Susan G.
Publication year - 2004
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704005289
Subject(s) - mathematics , measure (data warehouse) , combinatorics , alexander polynomial , integer (computer science) , infinity , zero (linguistics) , polynomial , homology (biology) , julia set , discrete mathematics , mathematical analysis , knot (papermaking) , computer science , knot theory , amino acid , biochemistry , chemistry , database , engineering , programming language , linguistics , philosophy , chemical engineering
Let l be an oriented link of d components in a homology 3‐sphere. For any nonnegative integer q , let l ( q ) be the link of d −1 components obtained from l by performing 1/ q surgery on its d th component l d . The Mahler measure of the multivariable Alexander polynomial Δ l(q) converges to the Mahler measure of Δ l as q goes to infinity, provided that l d has nonzero linking number with some other component. If l d has zero linking number with each of the other components, then the Mahler measure of Δ l ( q ) has a well defined but different limiting behavior. Examples are given of links l such that the Mahler measure of Δ l is small. Possible connections with hyperbolic volume are discussed.