Papers › Perturbed Traceless SU(2) Character Varieties of Tangle Sums
Perturbed Traceless SU(2) Character Varieties of Tangle Sums
Kai Smith
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If a link L can be decomposed into the union of two tangles T∪_(S²) S along a 2-sphere intersecting L in 4 points, then the intersections of perturbed traceless SU(2) character varieties of tangles in a space called the pillowcase form a set of generators for Kronheimer and Mrowka's reduced singular instanton homology, I^♮. It is conjectured by Cazassus, Herald, Kirk, and Kotelskiy that with the addition of bounding cochains, the differential of I^♮ can be recovered from these Lagrangians as well. This article gives a method to compute the perturbed character variety for a large class of tangles using cut-and-paste methods. In particular, given two tangles, T and S, Conway defines the tangle sum T+S. Given the character varieties of T and S, we show how to construct the perturbed character variety of T+S. This is done by first studying the perturbed character variety of a certain tangle C₃ properly embedded in S³ with 3 balls removed. Using these results, we prove a nontriviality result for the bounding cochains in the conjecture of Cazassus, Herald, Kirk, and Kotelskiy.
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