AI 中文总结
本文利用三维量子迹映射为长度猜想提供证据，证明其对无限族扭结一阶成立，并提出该映射对Dehn填充具自然行为的猜想。
AI 中文摘要
利用新开发的三维量子迹映射，为长度猜想提供更多证据，该猜想是全阶体积猜想的细化版本，纳入了扭结补的skein模中额外链环的插入。特别地，证明该长度猜想对无限族扭结在一阶成立，过程中提出三维量子迹映射对Dehn填充具有自然行为的猜想。
英文摘要
Using the newly developed $3$d quantum trace map, we compile more evidence for the length conjecture, a refinement of the all-order volume conjecture that incorporates insertions of additional links in the skein module of the knot complement. In particular, we prove that the length conjecture holds up to first order for an infinite family of twist knots. Along the way, we form a conjecture that the $3$d quantum trace map behaves naturally with respect to Dehn filling.
Comments34 pages
