Interactions of geometry and knot theory

这个项目旨在利用双曲几何和克莱因群的最新突破，将几何与纽结联系起来，纽结是微生物学、化学、物理和数学中出现的数学对象。纽结通常是通过它们周围的空间来研究的，被称为纽结互补。结的补码分解成几何碎片，而最常见的几何是双曲线，这完全决定了一个结。然而，如何从经典描述中获得关于纽结的双曲几何的信息是未知的。该项目将通过发现能够对即使是极其复杂的结进行分类的结果来获得信息，并可能影响数学和其他领域。

This project aims to use recent breakthroughs in hyperbolic geometry and Kleinian groups to relate geometry to knots which are mathematical objects arising in microbiology, chemistry, physics, and mathematics. Knots are often studied via the space around them known as the knot complement. Knot complements decompose into geometric pieces, and the most common geometry is hyperbolic, which completely determines a knot. However, how to obtain information on the hyperbolic geometry of a knot from a classical description is unknown. This project will obtain information by uncovering results that would enable classification of even extremely complicated knots, and could affect mathematics and other fields.