Weekly Log
Week 1: I met with Dr. Beheshti to discuss potential research
ideas for the summer. We decided to focus on Legendrian Knot Theory and
its applications to DNA topology in order to build on the topics we had
discussed in the spring. Dr. Beheshti designed a series of lectures on
Legendrian Knot Theory, and I began reading several sets of online lecture notes
on the subject. We also found several articles that provided a readable
background.
Week 2: Dr. Beheshti gave her second lecture of the series.
We theorized about how Legendrian Knot Theory can apply to medical imaging,
specifically in visualizing DNA knots using electron microscopy. I did
some background research on the history of visualizing knotted polymers using
microscopy in order to get a sense of the types of experimental techniques that
were being used. We concluded that many of these procedures were
labor-intensive, and that coming up with a mathematical procedure to
lessen the need for laboratory imaging would be useful.
Week 3: Dr. Beheshti gave her third and final lecture, which
focused on Legendrian isotopies between knots. We discussed how these
might be useful in a biophysical context. In particular, we would like to
show that there is a nonempty set of DNA front projections that correspond to a
Legendrian knot.
Week 4: Several experimental results from the mid-1990s
indicated that DNA intrinsic bend in promoter regions can have an effect on gene
regulation. We began devising a physical model to describe molecular
forces both in local regions and over large regions. A primary goal is
incorporating terms that account for sequence context.
Week 5: Following a very helpful conversation with Dr. Wilma
Olson, we decided to focus our efforts on describing "pinching" in knots, as
well as determining whether a given DNA image can be reoriented to form a
Legendrian knot. We expect these ideas to form the crux of our future
work.
Week 6: We developed the idea of "pinching classes" and are
working on determining how these pinches affect the linking number, writhe, and
twist of a knot. Currently, we are working on making diagrams of all
possible pinching conformations for knots that contain 0 through 6 pinches.
This may prove useful in describing some biophysical events.
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