The correspondence induced on the pillowcase by the earring tangle

2020/10/09 by Guillem Cazassus, Christopher M. Herald, Cazassus, Guillem +5 · 2 citations
Arts and Humanities · Mathematics · #57K18 #57K31 #57R58 #81T13 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Medieval European Literature and History #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2010.04320

openalex publication_date 2020/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The earring tangle consists of four strands 4pt x [0,1] inside S² x [0,1] and one meridian around one of the strands. Equipping this tangle with a nontrivial SO(3) bundle, we show that its traceless SU(2) flat moduli space is topologically a smooth genus three surface. We also show that the restriction map from this surface to the traceless flat moduli space of the boundary of the earring tangle is a particular Lagrangian immersion into the product of two pillowcases. The latter computation suggests that figure eight bubbling -- a subtle degeneration phenomenon predicted by Bottman and Wehrheim -- appears in the context of traceless character varieties.

Cited by

• Traceless SU(2) characters and ℤ/4 instanton gradings for two-bridge and (3,n)-torus knots
• The instanton homology of the (-2,3,q) pretzel knots and computed bounding cochains in the pillowcase

Related

• An endomorphism on immersed curves in the pillowcase
• Perturbed Traceless SU(2) Character Varieties of Tangle Sums
• The pillowcase and traceless representations of knot groups II: a\n Lagrangian-Floer theory in the pillowcase
• Bordered theory for pillowcase homology
• Floer field theory for tangles