UBC Theses and Dissertations
Thinking of (1,1) knots using elastic bands on peg-boards and combed glazing on the mille-feuille Marian, Mihai
Despite the analytic underpinnings of Heegaard Floer theory and its refine- ment to knots, there is an interesting class of knots, the (1, 1) knots, which have the special property that their knot Floer homology can be computed na ̈ıvely, straight from the definition, using only combinatorial techniques. In this thesis, we survey (1,1)-knots, describe their knot Floer homology, and focus in particular on the landscape of the manifolds obtained by Dehn surgery on these knots. More precisely, J. Greene, S. Lewallen and F. Vafaee recently described a simple criterion for determining if a (1, 1) knot admits a nontrivial surgery to an L-space, using the orientation of the curves in a doubly pointed genus-1 Heegaard diagram for the knot. This character- ization is formally very similar to a characterization due to J. Hanselman, J. Rasmussen and L. Watson, using a graphical calculus they developed for working with the bordered Floer theory. We relate these two perspectives, by providing in the final chapter a novel proof of Greene et al.’s criterion using the graphical calculus, recently expanded by A. Kotelskiy, Watson and C. Zibrowius.
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