Algebraic & Geometric Topology

Epimorphisms and boundary slopes of $2$–bridge knots

Jim Hoste and Patrick D Shanahan

Full-text: Open access


In this article we study a partial ordering on knots in S3 where K1K2 if there is an epimorphism from the knot group of K1 onto the knot group of K2 which preserves peripheral structure. If K1 is a 2–bridge knot and K1K2, then it is known that K2 must also be 2–bridge. Furthermore, Ohtsuki, Riley and Sakuma give a construction which, for a given 2–bridge knot Kpq, produces infinitely many 2–bridge knots Kpq with KpqKpq. After characterizing all 2–bridge knots with 4 or less distinct boundary slopes, we use this to prove that in any such pair, Kpq is either a torus knot or has 5 or more distinct boundary slopes. We also prove that 2–bridge knots with exactly 3 distinct boundary slopes are minimal with respect to the partial ordering. This result provides some evidence for the conjecture that all pairs of 2–bridge knots with KpqKpq arise from the Ohtsuki–Riley–Sakuma construction.

Article information

Algebr. Geom. Topol., Volume 10, Number 2 (2010), 1221-1244.

Received: 8 February 2010
Revised: 4 May 2010
Accepted: 10 May 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

knot $2$–bridge boundary slope epimorphism


Hoste, Jim; Shanahan, Patrick D. Epimorphisms and boundary slopes of $2$–bridge knots. Algebr. Geom. Topol. 10 (2010), no. 2, 1221--1244. doi:10.2140/agt.2010.10.1221.

Export citation


  • M Boileau, S Boyer, On character varieties, sets of discrete characters, and non-zero degree maps
  • M Boileau, S Boyer, A Reid, S Wang, Simon's conjecture for $2$–bridge knots
  • M Boileau, J H Rubinstein, S Wang, Finiteness of $3$–manifolds associated with non-zero degree mappings
  • G Burde, H Zieschang, Knots, second edition, de Gruyter Studies in Math. 5, de Gruyter, Berlin (2003)
  • D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of $3$–manifolds, Invent. Math. 118 (1994) 47–84
  • F González-Acuña, A Ramírez, Two-bridge knots with property $Q$, Q. J. Math. 52 (2001) 447–454
  • A Hatcher, W Thurston, Incompressible surfaces in $2$–bridge knot complements, Invent. Math. 79 (1985) 225–246
  • J Hoste, P D Shanahan, A formula for the A–polynomial of twist knots, J. Knot Theory Ramifications 13 (2004) 193–209
  • J Hoste, P D Shanahan, Boundary slopes of $2$–bridge links determine the crossing number, Kobe J. Math. 24 (2007) 21–39
  • K Ichihara, S Mizushima, Crossing number and diameter of boundary slope set of Montesinos knot, Comm. Anal. Geom. 16 (2008) 565–589
  • D Johnson, C Livingston, Peripherally specified homomorphs of knot groups, Trans. Amer. Math. Soc. 311 (1989) 135–146
  • T Kitano, M Suzuki, A partial order in the knot table, Experiment. Math. 14 (2005) 385–390
  • T Kitano, M Suzuki, A partial order in the knot table. II, Acta Math. Sin. $($Engl. Ser.$)$ 24 (2008) 1801–1816
  • P B Kronheimer, T S Mrowka, Witten's conjecture and property P, Geom. Topol. 8 (2004) 295–310
  • M Macasieb, K Petersen, R Van Luijk, On character varieties of two-bridge knot groups
  • T W Mattman, G Maybrun, K Robinson, $2$–bridge knot boundary slopes: diameter and genus, Osaka J. Math. 45 (2008) 471–489
  • T Ohtsuki, Ideal points and incompressible surfaces in two-bridge knot complements, J. Math. Soc. Japan 46 (1994) 51–87
  • T Ohtsuki, R Riley, M Sakuma, Epimorphisms between $2$–bridge link groups, from: “The Zieschang Gedenkschrift”, (M Boileau, M Scharlemann, R Weidmann, editors), Geom. Topol. Monogr. 14, Geom. Topol. Publ., Coventry (2008) 417–450
  • R Riley, Nonabelian representations of $2$–bridge knot groups, Quart. J. Math. Oxford Ser. $(2)$ 35 (1984) 191–208
  • D S Silver, W Whitten, Knot group epimorphisms, J. Knot Theory Ramifications 15 (2006) 153–166