The pillowcase and perturbations of traceless representations of knot groups

Hedden, Matthew¹ ; Herald, Chris M² ; Kirk, Paul³

¹ Department of Mathematics, Michigan State University, A338 WH, East Lansing, MI 48824, USA
² Department of Mathematics, University of Nevada, 1664 N. Virginia Street, Reno, NV 89557, USA
³ Department of Mathematics, Indiana University, Rawles Hall, 831 East 3 Street, Bloomington, IN 47405, USA

Geometry & topology, Tome 18 (2014) no. 1, pp. 211-287.

Résumé

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 $SO\left(3\right)$ 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.

Détail

DOI : 10.2140/gt.2014.18.211

Classification : 57M27, 57R58, 57M25, 81T13
Keywords: pillowcase, holonomy perturbation, instanton, Floer homology, character variety, two bridge knot, torus knot

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