Divisibility of twisted Alexander polynomials and fibered knots

Teruaki Kitano[1]; Takayuki Morifuji[2]

  • [1] Department of Mathematical and Computing Sciences Tokyo Institute of Technology 2-12-1-W8-43 Oh-okayama, Meguro-ku Tokyo 152-8552, Japan
  • [2] Department of Mathematics Tokyo University of Agriculture and Technology 2-24-16 Naka-cho, Koganei Tokyo 184-8588, Japan

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

  • Volume: 4, Issue: 1, page 179-186
  • ISSN: 0391-173X

Abstract

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We prove that Wada’s twisted Alexander polynomial of a knot group associated to any nonabelian S L ( 2 , 𝔽 ) -representation is a polynomial. As a corollary, we show that it is always a monic polynomial of degree 4 g - 2 for a fibered knot of genus  g .

How to cite

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Kitano, Teruaki, and Morifuji, Takayuki. "Divisibility of twisted Alexander polynomials and fibered knots." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.1 (2005): 179-186. <http://eudml.org/doc/84553>.

@article{Kitano2005,
abstract = {We prove that Wada’s twisted Alexander polynomial of a knot group associated to any nonabelian $SL(2,\mathbb \{F\})$-representation is a polynomial. As a corollary, we show that it is always a monic polynomial of degree $4g-2$ for a fibered knot of genus $g$.},
affiliation = {Department of Mathematical and Computing Sciences Tokyo Institute of Technology 2-12-1-W8-43 Oh-okayama, Meguro-ku Tokyo 152-8552, Japan; Department of Mathematics Tokyo University of Agriculture and Technology 2-24-16 Naka-cho, Koganei Tokyo 184-8588, Japan},
author = {Kitano, Teruaki, Morifuji, Takayuki},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {179-186},
publisher = {Scuola Normale Superiore, Pisa},
title = {Divisibility of twisted Alexander polynomials and fibered knots},
url = {http://eudml.org/doc/84553},
volume = {4},
year = {2005},
}

TY - JOUR
AU - Kitano, Teruaki
AU - Morifuji, Takayuki
TI - Divisibility of twisted Alexander polynomials and fibered knots
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2005
PB - Scuola Normale Superiore, Pisa
VL - 4
IS - 1
SP - 179
EP - 186
AB - We prove that Wada’s twisted Alexander polynomial of a knot group associated to any nonabelian $SL(2,\mathbb {F})$-representation is a polynomial. As a corollary, we show that it is always a monic polynomial of degree $4g-2$ for a fibered knot of genus $g$.
LA - eng
UR - http://eudml.org/doc/84553
ER -

References

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  1. [1] J. C. Cha, Fibred knots and twisted Alexander invariants, Trans. Amer. Math. Soc. 355 (2003), 4187–4200. Zbl1028.57004MR1990582
  2. [2] G. de Rham, Introduction aux polynomes d’un nœud, Enseign. Math. 13 (1968), 187–194. Zbl0157.54803
  3. [3] H. Goda, T. Kitano and T. Morifuji, Reidemeister torsion, twisted Alexander polynomial and fibered knots, Comment. Math. Helv. 80 (2005), 51–61. Zbl1066.57008MR2130565
  4. [4] H. Goda and T. Morifuji, Twisted Alexander polynomial for S L ( 2 , ℂ ) -representations and fibered knots, C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003), 97–101. Zbl1061.57012MR2013157
  5. [5] B. Jiang and S. Wang, Twisted topological invariants associated with representations, In: “Topics in Knot Theory”, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 399, Kluwer Academic Publishers, Dordrecht, 1993, 211–227. Zbl0815.55001MR1257911
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  10. [10] K. Kodama, http://www.math.kobe-u.ac.jp/HOME/kodama/knot.html 
  11. [11] X. S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl. Ser.) 17 (2001), 361-380. Zbl0986.57003MR1852950
  12. [12] T. Morifuji, A twisted invariant for finitely presentable groups, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 143–145. Zbl0988.57008MR1801675
  13. [13] T. Morifuji, Twisted Alexander polynomial for the braid group, Bull. Austral. Math. Soc. 64 (2001), 1–13. Zbl1008.20028MR1848073
  14. [14] L. Neuwirth, “Knot Groups”, Annals of Mathematics Studies, No. 56, Princeton University Press, Princeton, N.J., 1965. Zbl0184.48903MR176462
  15. [15] R. Riley, Nonabelian representations of 2 -bridge knot groups, Quarterly J. Math. Oxford (2) 35 (1984), 191–208. Zbl0549.57005MR745421
  16. [16] M. Suzuki, Twisted Alexander polynomial for the Lawrence-Krammer representation, Bull. Austral. Math. Soc. 70 (2004), 67–71. Zbl1065.57013MR2079361
  17. [17] M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994), 241–256. Zbl0822.57006MR1273784

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