It has been known since the early 1980's that knot complements satisfy the geometrization conjecture. Thus the geometry of knot complements ought to be a useful tool in the study of knots. However, what is not known in general is how to relate diagrams of knots to their geometric structure, particularly hyperbolic knots. In the last few years, the PI has found bounds on certain geometric information of a knot complement based on a diagram, including bounds on cusp shapes and volumes for classes of knots. In this project, the PI will apply techniques in 3-manifold theory developed in the last few years to knot complements, to further our understanding of their geometric properties.
A closed loop lying in space may be knotted or unknotted. In mathematics, such a loop is called a knot. One goal of knot theory is to determine, based on a snapshot of that knot (a diagram), whether it can be unknotted without breaking the loop. A method of studying knots is to consider not the knot itself, but the 3-dimensional space obtained by removing the knot from the 3-sphere, called the knot complement. The PI will use recent advances in the study of 3-dimensional spaces, or 3-manifold theory, to study knots. This research has applications to such areas as string theory and the knotting of DNA.
