 

Daniel S. Silver and
Susan G. Williams
Lehmer's question, graph complexity growth and links
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print


Published: 
July 13, 2021. 
Keywords: 
Knots, periodic graphs, Mahler measure, Lehmer's question. 
Subject: 
05C10, 37B10, 57M25, 82B20. 


Abstract
Lehmer's question, an open question about the Mahler measure of monic integral polynomials, is shown to be equivalent to a question about the complexity growth rate of signed 1periodic graphs. If G is a dperiodic graph
(i.e. G has a cofinite free Z^{d}action by automorphisms), then a dvariable polynomial Δ_{G}
can be defined with Mahler measure equal to the logarithmic growth rate γ_{G} of a complexity defined
for the finite quotients of G.
A plane 1periodic graph determines a link via projection and the medial graph construction. The polynomial
Δ_{G} can be determined from the Alexander polynomial of the link. The complexity growth rate
γ_{G} of any dperiodic graph is at least log 2. An investigation of plane 1 and 2periodic graphs yields more connections with knot theory including work of A. Champanerkar and I. Kofman.


Acknowledgements
Both authors are grateful for the support of the Simons Foundation


Author information
Daniel S. Silver:
Department of Mathematics and Statistics
University of South Alabama
Mobile, AL 36688, USA
silver@southalabama.edu
Susan G. Williams:
Department of Mathematics and Statistics
University of South Alabama
Mobile, AL 36688, USA
swilliam@southalabama.edu

