This dissertation explores some relationships between knots, graphs, and Feynman diagrams (also known as chord diagrams). We do this by investigating three well-known polynomials, the Penrose polynomial, the Alexander polynomial, and the transition polynomial.
First, we study the Penrose polynomial for plane graphs. Given a plane graph, we construct homology groups whose Euler characteristic is its Penrose polynomial evaluated at an arbitrary integer. Our work is motivated by Khovanov's categorification of the Jones polynomial for knots, and the subsequent categorifications of the chromatic and Tutte polynomials for graphs, by various authors.
Next, we study the Alexander polynomial for knots. We establish a graded homology theory for link diagrams that yields the Alexander polynomial when taking the graded Euler characteristic. Our construction is motivated by the problem of finding a simpler approach to the combinatorial knot Floer homology of Manolescu, Ozsvath, and Sarkar.
Finally, we study the transition polynomial which was introduced by Jaeger for four-regular plane graphs. We define a transition polynomial for Feynman diagrams. Feynman diagrams naturally arise in molecular biology. In particular, the genus of a diagram plays an important role in RNA folding. We show our transition polynomial encodes the information necessary to calculate the genus of a Feynman diagram.
|Commitee:||Kidwell, Mark, Przytycki, Jozef, Schmitt, William, Shumakovitch, Alexander|
|School:||The George Washington University|
|School Location:||United States -- District of Columbia|
|Source:||DAI-B 69/07, Dissertation Abstracts International|
|Keywords:||Alexander polynomials, Clock moves, Feynman diagrams, Khovanov homology, Knots, Penrose polynomial, Transition polynomials|
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