Fibred knots and twisted Alexander invariants

Author:
Jae Choon Cha

Journal:
Trans. Amer. Math. Soc. **355** (2003), 4187-4200

MSC (2000):
Primary 57M25

DOI:
https://doi.org/10.1090/S0002-9947-03-03348-8

Published electronically:
June 24, 2003

MathSciNet review:
1990582

Full-text PDF

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.

**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**

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:
https://doi.org/10.1090/S0002-9947-03-03348-8

Keywords:
Fibred knots,
twisted Alexander invariants

Received by editor(s):
October 5, 2001

Received by editor(s) in revised form:
February 15, 2003

Published electronically:
June 24, 2003

Article copyright:
© Copyright 2003
American Mathematical Society