Research Statement November 2006
نویسنده
چکیده
Classically, the study of knots and links has proceeded topologically looking for features of knotted curves which depend only on their knot class. Recently there has been an increased interest in geometric knot theory, which attempts to measure geometric properties of a particular knotted curve and relate these to its knot type. Questions about the geometry of knots not only have intrinsic mathematical interest, but arise naturally from the physical sciences. For example biologists are interested in knotted strands of DNA, whose electrophoresis gel velocities seem related to notions of knot complexity, and in particular to ropelength. My research in geometric knot theory has several intertwined components. To date, I have looked at two topological features of knots, alternating quadrisecants and essential secants (see Section 2.1, Section 2.2). These are of interest in their own right as they are related to finite-type invariants and the fundamental group of a knot respectively. I have used these ideas to discover more details about the geometry of knots, giving new proofs of results about the total curvature and second hull of knotted curves (see Section 3.1). I also have results giving new insight into the optimal configuration of knots. These configurations arise from optimizing some functional on a knot class, like the ropelength of thick knots (see Section 3.2) or distortion (see Section 3.3).
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تاریخ انتشار 2006