No Access

Normalization of twisted Alexander invariants

Takahiro Kitayama

Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan

https://doi.org/10.1142/S0129167X15500779Cited by:6 (Source: Crossref)

Abstract

Twisted Alexander invariants of knots are well defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, and refine the duality theorem. We further obtain necessary conditions on the invariants for a knot to be fibered, and study behavior of the highest degrees of the invariants.

Keywords:

AMSC: 57M25, 57M05, 57Q10

References

J. C. Cha , Trans. Amer. Math. Soc. 355 , 4187 ( 2003 ) . Crossref, Web of Science, Google Scholar
J. DeBlois , S. Friedl and S. Vidussi , Michigan Math. J. 63 , 65 ( 2014 ) . Crossref, Web of Science, Google Scholar
J. Dubois, S. Friedl and W. Lück, Introduction to Modern Mathematics, Advanced Lectures in Mathematics 33 (International Press, 2015) pp. 143–170. Google Scholar
N. M. Dunfield , S. Friedl and N. Jackson , Exp. Math. 21 , 329 ( 2012 ) . Crossref, Web of Science, Google Scholar
S. Friedl and T. Kim , Topology 45 , 929 ( 2006 ) . Crossref, Web of Science, Google Scholar
S. Friedl , T. Kim and T. Kitayama , Indiana Univ. Math. J. 61 , 147 ( 2012 ) . Crossref, Web of Science, Google Scholar
S. Friedl and S. Vidussi , Bull. Lond. Math. Soc. 39 , 614 ( 2007 ) . Crossref, Web of Science, Google Scholar
S. Friedl and S. Vidussi , Banach Center Publ. 85 , 43 ( 2009 ) . Crossref, Google Scholar
S. Friedl and S. Vidussi, Ann. of Math. (2) 173(3), 1587 (2011). Crossref, Web of Science, Google Scholar
S. Friedl and S. Vidussi, Twisted Alexander polynomials and fibered 3-manifolds, Low-Dimensional and Symplectic Topology82, Proceedings of Symposia in Pure Mathematics (American Mathematical Society, 2011) pp. 111–130. Google Scholar
S. Friedl and S. Vidussi, The Mathematics of Knots, Contributions in Mathematical Computational Sciences 1 (Springer, 2011) pp. 45–94. Crossref, Google Scholar
S. Friedl and S. Vidussi , J. Eur. Math. Soc. (JEMS) 15 , 2027 ( 2013 ) . Crossref, Web of Science, Google Scholar
S. Friedl and S. Vidussi, The Thurston norm and twisted Alexander polynomials, to appear in J. Reine Angew. Math . Google Scholar
H. Goda , T. Kitano and T. Morifuji , Comment. Math. Helv. 80 , 51 ( 2005 ) . Crossref, Web of Science, Google Scholar
H. Goda and T. Morifuji , C. R. Math. Acad. Sci. Soc. R. Can. 25 , 97 ( 2003 ) . Google Scholar
J. Hillman , C. Livingston and S. Naik , Algebr. Geom. Topol. 6 , 145 ( 2006 ) . Crossref, Web of Science, Google Scholar
P. Kirk and C. Livingston , Topology 38 , 635 ( 1999 ) . Crossref, Web of Science, Google Scholar
T. Kitano , Pacific J. Math. 174 , 431 ( 1996 ) . Crossref, Web of Science, Google Scholar
T. Kitano and T. Morifuji , Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 11 , 395 ( 2012 ) . Google Scholar
E. Klassen , Trans. Amer. Math. Soc. 326 , 795 ( 1991 ) . Crossref, Web of Science, Google Scholar
X. S. Lin , Acta Math. Sin. (Engl. Ser.) 17 , 361 ( 2001 ) . Crossref, Web of Science, Google Scholar
J. Milnor , Ann. of Math. 76 , 137 ( 1962 ) . Crossref, Web of Science, Google Scholar
T. Morifuji, Handbook of Group Actions, Advanced Lectures in Mathematics 31 (International Press, 2015) pp. 527–576. Google Scholar
D. Silver and S. G. Williams , J. Knot Theory Ramifications 20 , 427 ( 2011 ) . Link, Web of Science, Google Scholar
V. Turaev , Introduction to Combinatorial Torsions , Lectures in Mathematics, ETH Zürich ( Springer , 2001 ) . Crossref, Google Scholar
V. Turaev , Torsions of 3-Manifolds , Progress in Mathematics 208 ( Birkhäuser-Verlag , 2002 ) . Crossref, Google Scholar
M. Wada , Topology 33 , 241 ( 1994 ) . Crossref, Web of Science, Google Scholar
F. Waldhausen , Ann. of Math. (2) 108 , 135 ( 1978 ) . Crossref, Web of Science, Google Scholar