|
Fibred knots and twisted Alexander invariants
Author(s):
Jae
Choon
Cha
Journal:
Trans. Amer. Math. Soc.
355
(2003),
4187-4200.
MSC (2000):
Primary 57M25
Posted:
June 24, 2003
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
We study the twisted Alexander invariants of fibred knots. We establish necessary conditions on the twisted Alexander invariants for a knot to be fibred, and develop a practical method to compute the twisted Alexander invariants from the homotopy type of a monodromy. It is illustrated that the twisted Alexander invariants carry more information on fibredness than the classical Alexander invariants, even for knots with trivial Alexander polynomials.
References:
-
- 1.
- S. Akbulut and R. Kirby, Branched covers of surfaces in
 -manifolds, Math. Ann. 252 (1979/80), no. 2, 111-131. MR 82j:57001 - 2.
- G. Burde and H. Zieschang, Knots, Walter de Gruyter & Co., Berlin, 1985. MR 87b:57004
- 3.
- T. D. Cochran, Noncommutative Knot Theory, arXiv:math.GT/0206258.
- 4.
- R. H. Crowell and R. H. Fox, Introduction to knot theory, Springer-Verlag, New York, 1977, Reprint of the 1963 original, Graduate Texts in Mathematics, No. 57. MR 56:3829
- 5.
- R. H. Fox, Free differential calculus. III. Subgroups, Ann. of Math. (2) 64 (1956), 407-419. MR 20:2374
- 6.
- C. Gordon, Knots whose branched cyclic coverings have periodic homology, Trans. Amer. Math. Soc. 168 (1972), 357-370. MR 45:4394
- 7.
- -, Some aspects of classical knot theory, Knot theory (Proc. Sem., Plans-sur-Bex, 1977), Springer, Berlin, 1978, pp. 1-60. MR 80f:57002
- 8.
- B. J. Jiang and S. C. Wang, Twisted topological invariants associated with representations, Topics in knot theory (Erzurum, 1992), Kluwer Acad. Publ., Dordrecht, 1993, pp. 211-227.
- 9.
- P. Kirk and C. Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology 38 (1999), no. 3, 635-661. MR 2000c:57010
- 10.
- -, Twisted knot polynomials: inversion, mutation and concordance, Topology 38 (1999), no. 3, 663-671. MR 2000c:57011
- 11.
- T. Kitano, Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math. 174 (1996), no. 2, 431-442. MR 97g:57007
- 12.
- J. P. Levine, A characterization of knot polynomials, Topology 4 (1965), 135-141. MR 31:5194
- 13.
- X.-S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sinica (Engl. Ser). 17 (2001), 361-380. MR 2003f:57018
- 14.
- R. Riley, Growth of order of homology of cyclic branched covers of knots, Bull. London Math. Soc. 22 (1990), no. 3, 287-297. MR 92g:57017
- 15.
- D. Rolfsen, Knots and links, Publish or Perish Inc., Berkeley, CA, 1976, Mathematics Lecture Series, No. 7; corrected reprint, 1990. MR 58:24236; MR 95c:57018
- 16.
- M. Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994), no. 2, 241-256. MR 95g:57021
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
57M25
Retrieve articles in all Journals with MSC
(2000):
57M25
Additional Information:
Jae
Choon
Cha
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Address at time of publication:
Information and Communication University, 119 Munjiro, Yuseong-gu, Daejeon 305-714, Korea
Email:
jccha@indiana.edu, jccha@icu.ac.kr
DOI:
10.1090/S0002-9947-03-03348-8
PII:
S 0002-9947(03)03348-8
Keywords:
Fibred knots,
twisted Alexander invariants
Received by editor(s):
October 5, 2001
Received by editor(s) in revised form:
February 15, 2003
Posted:
June 24, 2003
Copyright of article:
Copyright
2003,
American Mathematical Society
|
Comments: webmaster@ams.org