Traceless SU(2) representations of 2-stranded tangles

Abstract

Given a 2-stranded tangle in a $\ZZ/2$ homology ball, $T\subset Y$, we investigate the character variety $R(Y,T)$ of conjugacy classes of traceless SU(2) representations of ${\pi }_1(Y\setminus T)$. In particular we completely determine the subspace of binary dihedral representations, and identify all of $R(Y,T)$ for many tangles naturally associated to knots in S^3. Moreover, we determine the image of the restriction map from $R(T,Y)$ to the traceless SU(2) character variety of the 4-punctured 2-sphere (the {\it pillowcase}). We give examples to show this image can be non-linear in general, and show it is linear for tangles associated to pretzel knots.

31 pages, color figures

References in corpus (1)

• Symplectic topology of SU(2)-representation varieties and link homology, I: Symplectic braid action and the first Chern class

Cited by in corpus (5)

• A class of knots with simple $SU(2)$ representations
• The pillowcase and traceless representations of knot groups II: a Lagrangian-Floer theory in the pillowcase
• Holonomy perturbations in a cylinder, and regularity for traceless SU(2) character varieties of tangles
• On spectral sequences from Khovanov homology
• Link homology and equivariant gauge theory