The focus of this project is application of geometric techniques to study braid theory with the intention to prove new results in knots theory. In particular, using methods developed in her recent work, the principal investigator will study the relationship between the braid index, a classical knot invariant, and various geometric quantities such as writhes, HOMFLY polynomials, self-linking number, and Khovanov-Rozansky homology. The latter provides an especially new and potent tool to study problems in braid theory. The PI also intends to explore the possibility of using open book decomposition, contact and Legendrian surgeries to analyze negative-flype-braid- moves, which have been playing a crucial role for classification of transversal knots in contact 3-manifolds. A goal is to find a new transversal knot invariant sensible to negative-flype-moves.
Braiding appears unexpectedly in algebraic geometry, cryptography, dynamical systems, homotopy groups of spheres, operator algebras, and robotics. In addition, knot and link theory is broadly applicable in a variety of scientific disciplines including biology and medicine where, for example, protein folding mechanisms and DNA structure play a seminal role. The impact of the project will likely stimulate communication and collaboration between low-dimensional topology and other fields of mathematics, theoretical physics and biology.
