2013/05/26 by Yoshihiro Fukumoto, Paul Kirk, Juanita Pinzón‐Caicedo +1
Mathematics · #Algebraic structures and combinatorial models #Character (mathematics) #Combinatorics #Conjugacy class #Geometric and Algebraic Topology #Geometry #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Pure mathematics #Subspace topology #Tangle #Variety (cybernetics) #math.GT #msc:57M25 #msc:57M27 #msc:57R58 #msc:81T13
paper · pdf · doi:10.1017/s0305004116000360
31 pages, color figures
arxiv created 2013/05/26 · openalex publication_date 2016/06/03 · arxiv updated 2016/06/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Abstract Given a 2-stranded tangle T contained in a ℤ-homology ball Y , we investigate the character variety R ( Y, T ) of conjugacy classes of traceless SU (2) representations of π 1 ( Y 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 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.