题    目：Chen-Yang’s volume conjecture for twist knots with rational Dehn surgeries

主讲人：葛化彬 教授

单    位：中国人民大学

时    间：2024年8月16日 16:30

地    点：公司二楼会议室

摘    要：The Volume Conjecture of Kashaev-Murakami-Murakami predicts a precise relation between the asymptotics of the n-colored Jones polynomials of a knot L in S^3 and the hyperbolic volume of its complement. Several years ago, Chen-Yang proposed a new “volume conjecture” for hyperbolic 3-manifolds, which gives a deep relation between the quantum SU(2) invariant (the Reshetikhin–Turaev invariant), the hyperbolic volume and Chern-Simons invariant of the manifolds. Their conjecture was later refined to include the adjoint twisted Reidemeister torsion in the asymptotic expansion of the invariants. Recently, Chen-Yang’s volume conjectures have been proved for many examples by various groups.

In this report, we will show our progress on Chen-Yang’s volume conjecture for hyperbolic 3-manifolds obtained by rational surgeries along the twist knot. To be precise, let s be the twist number of the twist knot K_s, and let (p,q) be the coefficients of the Dehn-filling. There exists a constant M such that when the absolute values of p, q, and s are greater than M, the Chen-Yang’s volume conjecture holds true. This development provides important clues for understanding the general validity of the Chen-Yang’s volume conjecture. This is joint work with Yunpeng Meng, Chuwen Wang and Yuxuan Yang.

简    介：葛化彬，北京大学数学科学学院博士、北京国际数学研究中心博士后，现为中国人民大学数学学院教授，博士生导师。主要研究方向为几何拓扑，推广了柯西刚性定理和Thurston圆堆积理论，部分解决Thurston的“几何理想剖分”猜想、完全解决Cheeger-Tian、林芳华的正则性猜想，相关论文分别发表在Geom. Topol., Geom. Funct. Anal., Amer. J. Math., Adv. Math.等著名数学期刊。