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The pillowcase and perturbations of traceless representations of knot groups. (English) Zbl 1377.57016

Summary: We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO\((3)\) connections relevant to Kronheimer and Mrowka’s singular instanton knot homology nondegenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians to produce minimal generating sets for singular instanton knot homology. The (symplectic) manifold where this intersection occurs corresponds to the traceless character variety of the four-punctured 2-sphere, which we identify with the familiar pillowcase. We investigate the image in this pillowcase of the traceless representations of tangles obtained by removing a trivial tangle from 2-bridge knots and torus knots. Using this, we compute the singular instanton homology of a variety of torus knots. In many cases, our computations allow us to understand nontrivial differentials in the spectral sequence from Khovanov homology to singular instanton homology.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R58 Floer homology
57M25 Knots and links in the \(3\)-sphere (MSC2010)
81T13 Yang-Mills and other gauge theories in quantum field theory

Software:

KnotTheory
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Full Text: DOI arXiv

References:

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