A fundamental goal in knot theory is the classification of knots up to isotopy and up to concordance. The fundamental group of the knot complement is an important algebraic invariant, but is rarely abelian. Homology with local coefficients can be used to associate modules to the knot. Since modules are abelian groups, distinguishing them is still feasible. However, since these are often modules over noncommutative rings, some of the rich structure of the fundamental group is retained. These homology modules, as well as linking forms defined on them, have been used by myself and others to define isotopy invariants of knots and concordance invariants of knots. This project concerns using these noncommutative techniques in two areas of knot theory: knot concordance and knot Floer homology.
The goal of this project is to use non-commutative algebra to better understand knotted curves in 3- and 4-dimensional space. The study of knots has important applications in biology and physics. For example, DNA strands are naturally knotted, but must unknot in order to replicate. Many mathematicians have used algebra to better understand how curves can knot, however the type of algebra considered is usually commutative. By employing non-commutative algebras, we can get a more refined understanding for how curves can knot in 3-dimensional space and bound surfaces in 4-dimensional space.
