Name: | Nicholas McConnell |
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Email: | ncm78 (at) dimax.rutgers.edu |
Office: | N/A |
Home Institution: | Rutgers University |
Project: | Heegaard-Floer homology and knot theory |
Mentor: | Kristen Hendricks |
My project is tied around knot theory and Heegaard-Floer homology. It starts out involving some basic homological algebra, along with the core concept of knots in S^{3}.
The goal of the summer research project is to study the Heegaard-Floer homology of knots and related knot invariants, to get a good view of their topological distinctions.
This website is sponsored by NSF Grant DMS-2019396 and the Rutgers University Mathematics Department.
This week, I read Peter Ozsvath's text on Heegaard's decompositions, and studied the knot groups for certain knots in general, using the Mayer-Vietoris sequence. Then I read Ciprian Manolescu's paper on knot Floer homology to get a better idea of what things are like.
I got better knowledge of how Heegaard decompositions of S^{3} can be represented in a way to indicate a specific knot, and how to use these indications to get the complex CFK.
The figure-8 knot's Heegaard-Floer homology was covered, along with the Sarkar involution and the map \iota_{k}.
Further studies of the Heegaard-Floer homology, particularly of 10_{161}. Will start thinking about the pretzel knot P(-2,m,k) with m and k odd.
This week, we learned how to derive the CFK complex for the pretzel knot P(-2,m,k), and worked out \iota_{k} in a few special cases. This approaches the goal of the REU project.
We studied thin knots, and the specific properties they have (e.g., the lack of nonzero filtered chain homotopies).
This week, I looked at my mentor's paper on the subject. It covered the diagrams for a few example knots, such as the figure 8 knot. One can generally do similar things for the pretzel knot.
Our paper intends to be about general formulas for the homology of pretzel knots.
Focused on changing basis in certain examples, and proving the main lemmas for pretzel knots: The \iota_{k} exchanges the filtration gradings (i,j) and (j,i). Also, there is no nonzero filtered map (even if it does not need to be a chain map) which sends homological grading s to s+1, hence homotopic maps such as \iota_{k}^{2} and the Sarkar involution must be equal.
This was the final week of the REU, with the presentations Wednesday-Friday. Me and my co-worker Matthew Isaac have presented our project at 11 AM.