Interactions of geometry and knot theory

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