AI 中文总结
本文证明了Dimofte-Garoufalidis关于双曲粘合方程幂级数常数项等于伴随Reidemeister挠率的猜想，并将Turaev-Viro型不变量的单圈项与此挠率关联。
AI 中文摘要
受量子不变量渐近性研究的启发，Dimofte和Garoufalidis引入了一个与带尖点双曲3流形的合适理想三角剖分相关的幂级数。他们证明了该幂级数的常数项可以用Neumann-Zagier数据和理想四面体的复形状参数来表示，并猜想它等于伴随扭曲Reidemeister挠率。另一方面，在研究带尖点3流形的Turaev-Viro型不变量渐近性时，作者与Liu、Sun和Yang一起发现，其渐近展开的单圈项可以用Gram矩阵和理想四面体的装饰边长来表示。在本文中，我们证明了Dimofte和Garoufalidis的猜想，并将Turaev-Viro型不变量中出现的单圈项与伴随扭曲Reidemeister挠率联系起来。
英文摘要
Motivated by the study of asymptotics of quantum invariants, Dimofte and Garoufalidis introduced a power series associated to a suitable ideal triangulation of a cusped hyperbolic $3$-manifold. They proved that its constant term can be written in terms of Neumann-Zagier data and the complex shape parameters of ideal tetrahedra, and conjectured that it equals the adjoint twisted Reidemeister torsion. On the other hand, in the study of asymptotics of Turaev-Viro type invariants of cusped $ 3$-manifolds, the authors, together with Liu, Sun and Yang, found that the one-loop terms of their asymptotic expansions could be written in terms of Gram matrices and decorated edge lengths of ideal tetrahedra. In this paper, we prove the conjecture of Dimofte and Garoufalidis and relate the one-loop term appearing in the Turaev-Viro type invariant to the adjoint twisted Reidemeister torsion.
Comments32 pages,3 figures
