Algebraic & Geometric Topology

Epimorphisms between $2$–bridge knot groups and their crossing numbers

Masaaki Suzuki

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Suppose that there exists an epimorphism from the knot group of a 2–bridge knot  K onto that of another knot K. We study the relationship between their crossing numbers c(K) and c(K). More specifically, it is shown that c(K) is greater than or equal to 3c(K), and we estimate how many knot groups a 2–bridge knot group maps onto. Moreover, we formulate the generating function which determines the number of 2–bridge knot groups admitting epimorphisms onto the knot group of a given 2–bridge knot.

Article information

Algebr. Geom. Topol., Volume 17, Number 4 (2017), 2413-2428.

Received: 15 June 2016
Revised: 1 October 2016
Accepted: 3 December 2016
First available in Project Euclid: 16 November 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}
Secondary: 57M27: Invariants of knots and 3-manifolds

epimorphism $2$–bridge knot knot group crossing number


Suzuki, Masaaki. Epimorphisms between $2$–bridge knot groups and their crossing numbers. Algebr. Geom. Topol. 17 (2017), no. 4, 2413--2428. doi:10.2140/agt.2017.17.2413.

Export citation


  • I Agol, The classification of non-free $2$–parabolic generator Kleinian groups, talk at Budapest Bolyai conference (2002)
  • I Agol, Y Liu, Presentation length and Simon's conjecture, J. Amer. Math. Soc. 25 (2012) 151–187
  • S Aimi, D Lee, M Sakuma, Parabolic generating pairs of $2$–bridge link groups In preparation
  • M Boileau, S Boyer, A,W Reid, S Wang, Simon's conjecture for two-bridge knots, Comm. Anal. Geom. 18 (2010) 121–143
  • G Burde, H Zieschang, M Heusener, Knots, 3rd edition, De Gruyter Studies in Mathematics 5, De Gruyter, Berlin (2014)
  • J Cha, C Livingston, KnotInfo: table of knot invariants, electronic reference Available at \setbox0\makeatletter\@url {\unhbox0
  • J,C Cha, M Suzuki, Non-meridional epimorphisms of knot groups, Algebr. Geom. Topol. 16 (2016) 1135–1155
  • C Ernst, D,W Sumners, The growth of the number of prime knots, Math. Proc. Cambridge Philos. Soc. 102 (1987) 303–315
  • S,M Garrabrant, J Hoste, P,D Shanahan, Upper bounds in the Ohtsuki–Riley–Sakuma partial order on $2$–bridge knots, J. Knot Theory Ramifications 21 (2012) art. id. 1250084, 24 pages
  • K Horie, T Kitano, M Matsumoto, M Suzuki, A partial order on the set of prime knots with up to $11$ crossings, J. Knot Theory Ramifications 20 (2011) 275–303
  • L,H Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395–407
  • T Kitano, M Suzuki, A partial order in the knot table, Experiment. Math. 14 (2005) 385–390
  • T Kitano, M Suzuki, Twisted Alexander polynomials and a partial order on the set of prime knots, from “Groups, homotopy and configuration spaces” (N Iwase, T Kohno, R Levi, D Tamaki, J Wu, editors), Geom. Topol. Monogr. 13, Geom. Topol. Publ., Coventry (2008) 307–321
  • L,M Milne-Thomson, The calculus of finite differences, Macmillan, London (1951)
  • K Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987) 187–194
  • K Murasugi, Jones polynomials and classical conjectures in knot theory, II, Math. Proc. Cambridge Philos. Soc. 102 (1987) 317–318
  • K Murasugi, Knot theory and its applications, Birkhäuser, Boston (1996)
  • 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
  • D Rolfsen, Knots and links, Math. Lect. Ser. 7, Publish or Perish, Berkeley, CA (1976)
  • D,S Silver, W Whitten, Knot group epimorphisms, II, preprint (2008)
  • M,B Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987) 297–309