| In the past ten to fifteen years, the topological subfield of knot theory has been rejuvenated as applied mathematics. The main area involved in this applied renaissance has been biochemistry. The knotting of macromolecules, like deoxyribonucleic acid (DNA), has become a topic of research for mathematicians.; Several mathematical models for macromolecular knotting have been developed. One which is often used is the equilateral random model. Using this framework, a macromolecule consists of a sequence of points, called vertices, in Euclidean 3-space. We imagine joining these vertices with line segments, called edges. Each vertex represents a molecule in the macromolecule under consideration; each edge represents the bonding between such molecules.; As with much applied mathematics, an intuitively "obvious" fact led to a great deal of rigorous work and exploration. Biochemists observed that the longer a chain of molecules became the more likely that it would become entangled. This is called the Frisch-Wasserman-Delbruck (F-W-D) conjecture.; The F-W-D conjecture has been proven for each of the existing mathematical models. It is important to note that the approaches used in these various proofs are strikingly similar: each is predominantly local in scope. Each shows that with high probability there are many tiny knots contained in the edges of a sufficiently long molecule. The size of each knotted component is small when compared with the size ofthe edge in which it is contained.; There is another approach to proving the F-W-D conjecture: it is an approach which is more global in nature. In this dissertation, we prove that with high probability a macromolecule, viewed as an equilateral random polygon, is contained in and is an essential loop of a knotted solid torus. This allows us to view the molecule as a satellite knot of this knotted solid torus. Ergo, macromolecular knotting occurs with high probability on a global scale--on a scale proportional to the diameter of the molecule. |
