Low-dimensional topology is primarily the study of 3 and 4 dimensional manifolds.
Important to both is the study of knots and links, which are smooth embeddings of disjoint copies of the circle into the 3-sphere. The central question of knot theory is the classification of knots and links up to isotopy.
This question is algorithmically difficult, so knot theories turn their attention to finding invariants of knots/links which help distinguish knots while also telling us something useful about the geometry of the knot or link.
One such invariant is the Alexander polynomial which is formally the GCD of a certain ideal in a polynomial ring coming from a certain covering space of the link exterior.
This invariant is one of the oldest invariants of knots/links, and together with its categorifications (knot/link Floer homology) plays an important role in the classical and modern study of links.
One role is the connection of the degree of this polynomial (the Newton Polytope) and the complexity of surfaces which bound certain components of the link; i.e. the Thurston norm $\|\cdot\|_T$ of the link exterior. One facet of my current research is trying to discern relationships between the Thurston norm and properties of links (in particular via the Alexander polynomial).
To see more details about this project, see the following page.