Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T07:23:22.935Z Has data issue: false hasContentIssue false
  • English
  • Français

The Asymptotics of the Higher Dimensional Reidemeister Torsion for Exceptional Surgeries Along Twist Knots

Published online by Cambridge University Press:  20 November 2018

Anh T. Tran  and
Yoshikazu Yamaguchi
Anh T. Tran
Affiliation:
Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA, e-mail: [email protected]
Yoshikazu Yamaguchi
Affiliation:
Department of Mathematics, Akita University, 1-1 Tegata-Gakuenmachi, Akita, 010-8502, Japan, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible $\text{S}{{\text{L}}_{2}}(\mathbb{C})$-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We also give the set of the limits of the leading coeõcients in the higher dimensional Reidemeister torsion explicitly.

Keywords

57M2757M50Reidemeister torsiongraph manifoldasymptotic behaviorexceptional surgery
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 1 , 01 March 2018 , pp. 211 - 224
Copyright
Copyright © Canadian Mathematical Society 2018

References

[BW01] Brittenham, M. and Wu, Y.-Q., The classification of exceptional Dehn surgeries on 2-bridge knots. Comm. Anal. Geom. 9 (2001), 97–113. http://dx.doi.org/10.4310/CAC.2001.v9.n1.a4Google Scholar
[CT13] Clay, A. and Teragaito, M., Lefi-orderability and exceptional Dehn surgery on two-bridge knots. In: Geometry and topology down under, Contemp. Math., 597, American Mathematical Society, Providence, RI, 2013, pp. 225-233. http://dx.doi.Org/10.1090/conm/597/11760Google Scholar
[HS04] Hoste, J. and Shanahan, P. D., Aformulafor the A-polynomial of twist knots. J. Knot Theory Ramifications 13 (2004), no. 2, 193–209. http://dx.doi.Org/10.1142/S0218216504003081Google Scholar
[MFP14] Menal-Ferrer, P. and Porti, J., Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds. J. Topol. 7 (2014), 69–119. http://dx.doi.Org/10.1112/jtopol/jtt024Google Scholar
[Mil66] Milnor, J., Whitehead torsion. Bull. Amer. Math. Soc. 72 (1966), 358–426. http://dx.doi.org/10.1090/S0002-9904-1966-11484-2Google Scholar
[NagO7] Nagasato, F., Finiteness ofa section of the SL(2, C)-character variety of knot groups. Kobe J. Math. 24 (2007), 125–136.Google Scholar
[NY12] Nagasato, F. and Yamaguchi, Y., On the geometry ofthe slice of trace-free characters of a knot group. Math. Ann. 354 (2012), 967–1002. http://dx.doi.Org/10.1007/s00208-011-0754-0Google Scholar
[Pat95] Patton, R., Incompressible punctured tori in the complements of alternating knots. Math. Ann. 301 (1995), 1–22. http://dx.doi.org/10.1007/BF01446618Google Scholar
[Por] Porti, J., Reidemeister torsion, hyperbolic three-manifolds, and character varieties. arxiv:1 511.00400Google Scholar
[TerO3] Teragaito, M., Toroidal surgeries on hyperbolic knots. II. Asian J. Math. 7 (2003), 139–146. http://dx.doi.org/10.4310/AJM.2003.v7.n1.a9Google Scholar
[Terl3] Teragaito, M., Lefi-orderability and exceptional Dehn surgery on twist knots. Canad. Math. Bull. 56 (2013), 850–859. http://dx.doi.Org/10.4153/CMB-2O12-011-0Google Scholar
[TY] Tran, A. T. and Yamaguchi, Y., Higher dimensional twisted Alexanderpolynomials for metabelian representations. arxiv:1608.0552 5Google Scholar
[TurOl] Turaev, V., Introduction to combinatorial torsions. Lectures in Mathematics, Birkhäuser Verlag, Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8321-4Google Scholar
[YamO7] Yamaguchi, Y., Limit values ofthe non-acyclic Reidemeister torsionfor knots. Algebr. Geom. Topol. 7 (2007), 1485–1507. http://dx.doi.Org/10.2140/agt.2007.7.1485Google Scholar
[Yaml3] Yamaguchi, Y., On the twisted Alexanderpolynomial for metabelian representations into SL2(C). Topology Appl. 160 (2013), 1760–1772. http://dx.doi.Org/10.1016/j.topol.2013.07.006Google Scholar
[Yam] Yamaguchi, Y., A surgery formula for the asymptotics of the higher dimensional Reidemeister torsion and Seifertfibered Spaces. Indiana Univ. Math. J. 66 (2017), 463–493. http://dx.doi.Org/10.1512/iumj.2O17.66.6012Google Scholar