Authors introduce the HZ character expansion of the HOMFLY-PT polynomial, identify hook diagrams for factorisability, and construct an infinite family of HZ-factorisable hyperbolic knots via full, partial-full, and Jucys-Murphy twists, plus a decomposition conjecture proven for three strands.
Lehmer's Problem, McKay's Correspondence, and $2,3,7$
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abstract
This paper addresses a long standing open problem due to Lehmer in which the triple 2,3,7 plays a notable role. Lehmer's problem asks whether there is a gap between 1 and the next smallest algebraic integer with respect to Mahler measure. The question has been studied in a wide range of contexts including number theory, ergodic theory, hyperbolic geometry, and knot theory; and relates to basic questions such as describing the distribution of heights of algebraic integers, and of lengths of geodesics on arithmetic surfaces. This paper focuses on the role of Coxeter systems in Lehmer's problem. The analysis also leads to a topological version of McKay's correspondence.
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The HZ character expansion and a hyperbolic extension of torus knots
Authors introduce the HZ character expansion of the HOMFLY-PT polynomial, identify hook diagrams for factorisability, and construct an infinite family of HZ-factorisable hyperbolic knots via full, partial-full, and Jucys-Murphy twists, plus a decomposition conjecture proven for three strands.