Random Knotting and Lattice Paths

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Knot theory has played an important role in molecular biology since entanglements in DNA molecules can affect cellular processes such as replication and transcription. Much of my research is aimed at understanding the phenomenon of random knotting. I am especially interested in lattice models where combinatorial methods can be used and my aims are to understand the relative frequency of occurrence of different knot types, and to derive bounds on the knot probability. Modern micromanipulation techniques mean that individual polymer molecules can be subjected to a force. My research is designed to construct and solve lattice models that help to understand and interpret these measurements and, more generally, how polymers respond to applied forces, especially when a polymer is pulled from a surface or pulled from a compact to an extended form. The primary aim is to study exactly solvable models, using combinatorial methods, but I also use numerical methods such as Monte Carlo techniques where rigorous solutions are beyond reach.