Legendrian Knots

Posted May 28th, 2010 at 10:38 am.

Sumana Shrestha
Mentor: Dr. Paul Melvin

Ever since it was discovered that the strands of DNA could be knotted, which impede DNA replication and reduce transcription, statistical studies of knots have been more important for molecular biology than ever before. [1] We use knot invariants to differentiate between two knots. Legendrian knots are special type of knots. They satisfy a geometrical condition imposed by a contact structure. There are many open-ended questions to be discussed about Legendrian knots. For example when do augmentation exist in the computation of contact homology? If they exist then what feature of the knot determines the number of augmentations? Will all the augmentation results in same linearized homology? Is the total dimension of the linearized homology, when they exist, a topological invariant and what is its significance? The answers to these questions may yield properties of Legendrian knots and will build upon the existing knowledge of these knots.

My research with Professor Paul Melvin will try to answer some of the questions posed above by finding patterns among the knots tabulated by Mark Branson in his paper “Tabulation of the Legendrain knots by Ambient Isotopy.” The first part of my research has dealt with finding braid words for these knots to use as input for a computer program written by Professor Melvin that computes some of the invariants mentioned above. I plan to tabulate my findings in an effort to discover patterns that might lead to new theoretical results

[1] http://www.nordita.dk/~metz/dnaknots.html

Filed under: 2004,Melvin, Dr. Paul,Shrestha, Sumana by Ann Dixon

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