Algebraic & Geometric Topology

Non-meridional epimorphisms of knot groups

Jae Choon Cha and Masaaki Suzuki

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

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

Abstract

In the study of knot group epimorphisms, the existence of an epimorphism between two given knot groups is mostly (if not always) shown by giving an epimorphism which preserves meridians. A natural question arises: is there an epimorphism preserving meridians whenever a knot group is a homomorphic image of another? We answer in the negative by presenting infinitely many pairs of prime knot groups (G,G) such that G is a homomorphic image of G but no epimorphism of G onto G preserves meridians.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 2 (2016), 1135-1155.

Dates
Received: 16 May 2015
Revised: 1 June 2015
Accepted: 10 June 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841162

Digital Object Identifier
doi:10.2140/agt.2016.16.1135

Mathematical Reviews number (MathSciNet)
MR3493417

Zentralblatt MATH identifier
1345.57007

Subjects
Primary: 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 20J05: Homological methods in group theory 57M05: Fundamental group, presentations, free differential calculus 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
knot groups epimorphisms meridians twisted Alexander polynomials

Citation

Cha, Jae Choon; Suzuki, Masaaki. Non-meridional epimorphisms of knot groups. Algebr. Geom. Topol. 16 (2016), no. 2, 1135--1155. doi:10.2140/agt.2016.16.1135. https://projecteuclid.org/euclid.agt/1510841162


Export citation

References

  • I Agol, Y Liu, Presentation length and Simon's conjecture, J. Amer. Math. Soc. 25 (2012) 151–187
  • J,C Cha, C Livingston, KnotInfo: table of knot invariants, electronic resource (2015) Available at \setbox0\makeatletter\@url http://www.indiana.edu/~knotinfo {\unhbox0
  • J,C Cha, M Suzuki, Table of twisted Alexander polynomials of $J_{-1}$ over $\mathrm{SL}(2,\mathbb{F}_7)$ (2016) online supplement Available at \setbox0\makeatletter\@url http://tbd {\unhbox0
  • M Culler, N,M Dunfield, J,R Weeks, SnapPy, a computer program for studying the geometry and topology of $3$–manifolds (2014) Available at \setbox0\makeatletter\@url http://snappy.computop.org/ {\unhbox0
  • R,H Fox, Free differential calculus, I: Derivation in the free group ring, Ann. of Math. 57 (1953) 547–560
  • F González-Acuña, Homomorphs of knot groups, Ann. of Math. 102 (1975) 373–377
  • F González-Acuña, A Ramírez, Two-bridge knots with property $Q$, Q. J. Math. 52 (2001) 447–454
  • F González-Acuña, A Ramírez, Epimorphisms of knot groups onto free products, Topology 42 (2003) 1205–1227
  • 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
  • J Hoste, P,D Shanahan, Trace fields of twist knots, J. Knot Theory Ramifications 10 (2001) 625–639
  • J Hoste, P,D Shanahan, Epimorphisms and boundary slopes of $2$–bridge knots, Algebr. Geom. Topol. 10 (2010) 1221–1244
  • D Johnson, Homomorphs of knot groups, Proc. Amer. Math. Soc. 78 (1980) 135–138
  • 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
  • T Kitano, M Suzuki, M Wada, Twisted Alexander polynomials and surjectivity of a group homomorphism, Alg. Geom. Topol. 5 (2005) 1315–1324 Correction in Alg. Geom. Topol. 11 (2011) 2937–2939
  • D Lee, M Sakuma, Epimorphisms between $2$–bridge link groups: homotopically trivial simple loops on $2$–bridge spheres, Proc. Lond. Math. Soc. 104 (2012) 359–386
  • 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 (2008) 417–450
  • R Riley, Parabolic representations of knot groups, I, Proc. London Math. Soc. 24 (1972) 217–242
  • D,S Silver, W Whitten, Knot group epimorphisms, J. Knot Theory Ramifications 15 (2006) 153–166
  • D,S Silver, W Whitten, Knot group epimorphisms, II, preprint (2008)
  • M Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994) 241–256

Supplemental materials

  • Table of twisted Alexander polynomials of the knot $J_{-1}$.