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Kai Smith - Character Varieties of Tangles and Singular Instanton Homology

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Jan 27, 12 pm ET: Kai Smith (Indiana University) - Character Varieties of Tangles and Singular Instanton Homology
Abstract: Hedden, Herald, and Kirk initiated a program to study knots using Lagrangian Floer Theory. Given a certain decomposition of the knot and appropriate perturbations, one can use character varieties to obtain a pair of Lagrangians in a space called the pillowcase. The intersections of these Lagrangians can be used to give bounds on the singular instanton homology of the knot. My work shows how to find these Lagrangians for tangle sums. Using this to compute new examples, I show that the Lagrangian Floer homology alone is not a knot invariant, suggesting that deriving a knot invariant from these Lagrangians will require more sophisticated methods.
Abstract: Hedden, Herald, and Kirk initiated a program to study knots using Lagrangian Floer Theory. Given a certain decomposition of the knot and appropriate perturbations, one can use character varieties to obtain a pair of Lagrangians in a space called the pillowcase. The intersections of these Lagrangians can be used to give bounds on the singular instanton homology of the knot. My work shows how to find these Lagrangians for tangle sums. Using this to compute new examples, I show that the Lagrangian Floer homology alone is not a knot invariant, suggesting that deriving a knot invariant from these Lagrangians will require more sophisticated methods.