发表机构
Osaka Metropolitan University（大阪公立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探讨纽结Q多项式的马勒测度与根分布，证明扭转下Q多项式马勒测度收敛、多数根趋近[-2,2]，对比三类多项式根的差异，给出仅含非零实根的2桥纽结无限族并提出相关猜想。
AI 中文摘要
我们研究纽结的Q多项式的根与马勒测度。首先考虑通过在一对平行股线中添加扭转得到的纽结，证明当扭转次数增加时，变换后Q多项式的马勒测度收敛；还表明，除有限个（数量一致有界）不同根外，Q多项式的所有根都趋近于实区间[-2,2]，该行为与琼斯多项式在扭转下的根不同。对素纽结的数值实验使我们对交错纽结变换后Q多项式的根提出猜想；最后，我们比较亚历山大多项式、琼斯多项式和Q多项式的实根与单位圆根，给出无限族2桥纽结，其Q多项式仅含非零实根。
英文摘要
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.
Comments14 pages, v2: Removed a conjecture, comments are welcome!
