A spherical CR structure on the complement of the figure eight knot with discrete holonomy

A 1-parameter family of spherical CR uniformizations of the figure eight knot complement

We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot complement. Finally, we construct an explicit 1-parameter family of deformations of the boundary unipotent holonomy group such that the boundary holonomy is twist-parabolic. For small values of the twist of these parabolic elements, this produces a 1-parameter family of pairwise non-…

A formally Kahler structure on a knot space of a G2-manifold

A knot space in a manifold M is a space of oriented immersions from a circle S^1 to M up to Diff(S^1). Brylinski has shown that a knot space of a Riemannian threefold is formally Kahler. We prove that a space of knots in a holonomy G2 manifold is formally Kahler.

计算机辅助的射影刚性研究

Computer Assisted Projective Rigidity

本文通过计算机辅助证明八字纽结补空间中约两千次远离理想点的Dehn手术是无穷小射影刚性的，补充了Heusener与Porti的相关结果，并给出了其在PSO(3,1)中的显式有理表示。

Self-contact Sets for 50 Tightly Knotted and Linked Tubes

We report on new numerical computations of the set of self-contacts in tightly knotted tubes of uniform circular cross-section. Such contact sets have been obtained before for the trefoil and figure eight knots by simulated annealing -- we use constrained gradient-descent to provide new self-contact sets for those and 48 other knot and link types. The minimum length of all unit diameter tubes in a given knot or link type is called the ropelength of that class of curves. Our computations yield improved upper bounds for the ropelength of all knots and links with 9 or fewer crossings except the t…

基于约束梯度下降的绳结收紧方法

Knot Tightening By Constrained Gradient Descent

该研究提出约束梯度下降法求解固定厚度下近似长度最小的打结多边形，推导了相关理论准则，得到379个覆盖指定交叉数范围的素纽结与链环的绳长结果及自接触图谱，提升了纽结绳长精度并首次发表链环绳长数据。