1975/03/01 by Robert W. Riley · 1 citation
Mathematics · Computer Science · #Geometric and Algebraic Topology #semigroups and automata theory #Mathematics #Knot (papermaking) #Section (typography) #Group (periodic table) #PSL #Combinatorics #Pure mathematics #Fundamental group #Transformation group #Physics #Computer science #Quantum mechanics
paper · doi:10.1017/s0305004100051094
openalex publication_date 1975/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21
When k is a 2-bridge knot with group π K , there are parabolic representations (p-reps) θ: π K → PSL( ): = PSL(2, ). The most obvious problem that this suggests is the determination of a presentation for an image group π K θ. We shall settle the easiest outstanding case in section 2 below, viz. k the figure-eight knot 4 1 , which has the 2-bridge normal form (5, 3). We shall prove that the (two equivalent) p -reps θ for this knot are isomorphisms of π K on π K θ. Furthermore, the universal covering space of S 3 k can be realized as Poincaré's upper half space 3 , and π K θ is a group of hyperbolic isometries of 3 which is also the deck transformation group of the covering 3 → S 3 k . The group π K θ is a subgroup of two closely related groups that we study in section 3. We shall give fundamental domains, presentations, and other information for all these groups.