On the Mahler measure and root distribution of the Q-polynomial of links
Abstract
We study the roots and the Mahler measure of the Q-polynomial of links. We first consider links obtained by adding twists to a pair of parallel strands. We prove that the Mahler measure of the transformed Q-polynomial converges as the number of twists increases. We also show that all but a uniformly bounded number of distinct roots of the Q-polynomial approach the real interval [−2,2]. This behavior is different from that of the roots of the Jones polynomial under twisting. Numerical experiments on prime knots lead us to a conjecture about the roots of the transformed Q-polynomial of alternating knots. Finally, we compare real and unit-circle roots of the Alexander, Jones, and Q-polynomials, and give an infinite family of $2$-bridge links whose Q-polynomials have only real nonzero roots.
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BibTeX
@misc{shoji2026mahler,
title = {{On the Mahler measure and root distribution of the $Q$-polynomial of links}},
author = {Kotaro Shoji},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05200},
eprint = {2609.05200},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05200},
}
RIS
TY - UNPB
TI - On the Mahler measure and root distribution of the Q-polynomial of links
AU - Kotaro Shoji
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.05200
UR - https://arxiv.org/abs/2609.05200
AB - We study the roots and the Mahler measure of the Q-polynomial of links. We first consider links obtained by adding twists to a pair of parallel strands. We prove that the Mahler measure of the transformed Q-polynomial converges as the number of twists increases. We also show that all but a uniformly bounded number of distinct roots of the Q-polynomial approach the real interval [−2,2]. This behavior is different from that of the roots of the Jones polynomial under twisting. Numerical experiments on prime knots lead us to a conjecture about the roots of the transformed Q-polynomial of alternating knots. Finally, we compare real and unit-circle roots of the Alexander, Jones, and Q-polynomials, and give an infinite family of $2$-bridge links whose Q-polynomials have only real nonzero roots.
ER -