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The pillowcase and perturbations of traceless representations of knot groups

Matthew Hedden, Christopher M Herald and Paul Kirk

Geometry & Topology 18 (2014) 211–287

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.

pillowcase, holonomy perturbation, instanton, Floer homology, character variety, two bridge knot, torus knot
Mathematical Subject Classification 2010
Primary: 57M27, 57R58, 57M25
Secondary: 81T13
Received: 5 January 2013
Accepted: 13 July 2013
Published: 23 January 2014
Proposed: Ronald Fintushel
Seconded: Simon Donaldson, Ronald Stern
Matthew Hedden
Department of Mathematics
Michigan State University
A338 WH
East Lansing, MI 48824
Christopher M Herald
Department of Mathematics
University of Nevada
1664 N. Virginia Street
Reno, NV 89557
Paul Kirk
Department of Mathematics
Indiana University
Rawles Hall
831 East 3 Street
Bloomington, IN 47405