Journal of Symplectic Geometry

Volume 16 (2018)

Number 3

The pillowcase and traceless representations of knot groups II: a Lagrangian–Floer theory in the pillowcase

Pages: 721 – 815

DOI: https://dx.doi.org/10.4310/JSG.2018.v16.n3.a5

Authors

Matthew Hedden (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

Christopher M. Herald (Department of Mathematics and Statistics, University of Nevada, Reno, Nv., U.S.A.)

Paul Kirk (Department of Mathematics, Indiana University, Bloomington, In., U.S.A.)

Abstract

We define an elementary relatively $\mathbb{Z}/4$ graded Lagrangian–Floer chain complex for restricted immersions of compact $1$-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless $SU(2)$ character varieties of $2$-tangle decompositions of knots. Calculations for torus knots are explained in terms of pictures in the punctured plane. The relation to the reduced instanton homology of knots is explored.

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The first author gratefully acknowledges support from NSF grant DMS-0906258, NSF CAREER grant DMS-1150872, and an Alfred P. Sloan Research Fellowship. The third author gratefully acknowledges support from NSF grant DMS-1007196 and Simons Foundation Collaboration Grant 278714.

The authors thank Juanita Pinzon-Caicedo and Yoshihiro Fukumoto for helpful discussions and assistance with writing the computer programs to produce the data and figures used in Section 11. They also thank Nikolai Saveliev and Matt Hogancamp for providing very useful insights, and Paul Seidel for pointing us to some influential references.

Received 28 June 2015

Accepted 8 December 2017

Published 26 November 2018