AI 中文总结
本文研究含本质Conway球面的纽结外部的$SL_2(\boldsymbol{C})$特征簇的实理想点，结合相关结果验证了部分纽结大斜率Dehn填充的$L$空间猜想，还对三类纽结族及28个普查外部完成了相关计算验证。
AI 中文摘要
我们研究包含本质Conway球面的纽结外部的$SL_2(\boldsymbol{C})$特征簇的实理想点。在两个互补缠结上明确的形变理论假设下，我们证明本质Conway球面可通过带实理想点的Culler-Shalen理论检测；这类理想点随后可在两侧形变为$SL_2(\boldsymbol{R})$表示的弧。接着我们计算这些弧上表示的渐近行为，并计算其提升至$\boldsymbol{\tilde{PSL}}_2(\boldsymbol{R})$表示的平移数。结合Gao的一个结果，我们得出：对于某些具有本质Conway球面的纽结，所有足够大的正、负有理填充都具有左可序基本群。这验证了文献中此前未知的纽结大斜率Dehn填充的$L$空间猜想。我们验证了由环面平凡缠结和扭平凡缠结组装成的三个无限纽结族的判据，随后计数得到的实理想点，确定所有构造分支的纵向平移数，并精确识别零平移弧。表\ ef{tab:instances}中列出的28个不同已验证普查外部的计算结果与已认证分支一致。
英文摘要
We study real ideal points of $SL_2(\mathbb C)$-character varieties of knot exteriors containing essential Conway spheres. Under explicit deformation-theoretic hypotheses on the two complementary tangles, we show that essential Conway spheres are detected via Culler-Shalen theory with real ideal points; such ideal points are then deformed into arcs of $SL_2(\mathbb R)$ representations on both sides. We then compute the asymptotic behavior of the representations on these arcs and compute the translation numbers of their lifts to $\widetilde{PSL}_2(\mathbb R)$ representations. Combining this with a result of Gao, we conclude that for certain knots with essential Conway spheres, all sufficiently large positive and negative rational fillings have left-orderable fundamental group. This verifies the $L$-space conjecture for large-slope Dehn fillings of knots which were previously unknown in the literature. We verify the criteria for three infinite families of knots assembled from torus-trivial and twist-trivial tangles, then count the resulting real ideal points, determine the longitudinal translation numbers of all constructed branches, and identify exactly the zero-translation arcs. Computations for the $28$ distinct verified census exteriors listed in Table \ref{tab:instances} agree with the certified branches.
Comments69 pages; 16 figures; comments welcome!
