2021/09/22 by Wang, Yi-Sheng · 2 citations
#57K10 #57M15 #FOS: Mathematics #Geometric Topology (math.GT) #Primary 57K12 #Secondary 57K30
paper · doi:10.48550/arxiv.2109.10609
A recent result of Funayoshi-Koda shows that a handlebody-knot of genus two has a finite symmetry group if and only if it is hyperbolic -- the exterior admits a hyperbolic structure with totally geodesic boundary -- or irreducible, atoroidal, cylindrical -- the exterior contains no essential disks or tori but contains an essential annulus. Based on the Koda-Ozawa classification theorem, essential annuli in an irreducible, atoroidal handlebody-knots of genus two are classified into four classes: type 2, type 3-2, type 3-3 and type 4-1. We show that under mild condition most genus two cylindrical handlebody-knot exteriors contain no essential disks or tori, and when a type 3-3 annulus exists, it is often unique up to isotopy; a classification result for symmetry groups of such cylindrical handlebody-knots is also obtained.