Geodesic
The L2–Alexander torsion is symmetric
Dubois, Jérôme1 ; Friedl, Stefan2 ; Lück, Wolfgang3
1Laboratoire de Mathématiques UMR 6620 – CNRS, Université Blaise Pascal, Campus des Cézeaux, BP 80026, 63171 Aubière, France
2Fakultät für Mathematik, Universität Regensburg, D-93053 Regensburg, Germany
3Mathematisches Institut, Rheinische Wilhelms-Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany
Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3599-3612
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We show that the L2–Alexander torsion of a 3–manifold is a symmetric function. This can be viewed as a generalization of the symmetry of the Alexander polynomial of a knot.

DOI : 10.2140/agt.2015.15.3599
Classification : 57M27, 57Q10
Keywords: $L^2$–Alexander torsion, duality, Thurston norm, knot genus

Dubois, Jérôme  1   ; Friedl, Stefan  2   ; Lück, Wolfgang  3

1 Laboratoire de Mathématiques UMR 6620 – CNRS, Université Blaise Pascal, Campus des Cézeaux, BP 80026, 63171 Aubière, France
2 Fakultät für Mathematik, Universität Regensburg, D-93053 Regensburg, Germany
3 Mathematisches Institut, Rheinische Wilhelms-Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany 
@article{10_2140_agt_2015_15_3599,
     author = {Dubois, J\'er\^ome and Friedl, Stefan and L\"uck, Wolfgang},
     title = {The {L2{\textendash}Alexander} torsion is symmetric},
     journal = {Algebraic and Geometric Topology},
     pages = {3599--3612},
     year = {2015},
     volume = {15},
     number = {6},
     doi = {10.2140/agt.2015.15.3599},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.3599/}
}
TY  - JOUR
AU  - Dubois, Jérôme
AU  - Friedl, Stefan
AU  - Lück, Wolfgang
TI  - The L2–Alexander torsion is symmetric
JO  - Algebraic and Geometric Topology
PY  - 2015
SP  - 3599
EP  - 3612
VL  - 15
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.3599/
DO  - 10.2140/agt.2015.15.3599
ID  - 10_2140_agt_2015_15_3599
ER  - 
%0 Journal Article
%A Dubois, Jérôme
%A Friedl, Stefan
%A Lück, Wolfgang
%T The L2–Alexander torsion is symmetric
%J Algebraic and Geometric Topology
%D 2015
%P 3599-3612
%V 15
%N 6
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.3599/
%R 10.2140/agt.2015.15.3599
%F 10_2140_agt_2015_15_3599
Dubois, Jérôme; Friedl, Stefan; Lück, Wolfgang. The L2–Alexander torsion is symmetric. Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3599-3612. doi: 10.2140/agt.2015.15.3599

[1] F Ben Aribi, The $L^2$–Alexander invariant detects the unknot, C. R. Math. Acad. Sci. Paris 351 (2013) 215

[2] F Ben Aribi, The $L^2$–Alexander invariant detects the unknot, C. R. Math. Acad. Sci. Paris 351 (2013) 215

[3] T D Cochran, Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004) 347

[4] J Dubois, S Friedl, W Lück, The $L^2$–Alexander torsion of $3$–manifolds, preprint (2014)

[5] J Dubois, S Friedl, W Lück, Three flavors of twisted knot invariants, preprint (2014)

[6] J Dubois, C Wegner, $L^2$–Alexander invariant for torus knots, C. R. Math. Acad. Sci. Paris 348 (2010) 1185

[7] J Dubois, C Wegner, Weighted $L^2$–invariants and applications to knot theory, Commun. Contemp. Math. 17 (2015) 1450010, 29

[8] G Elek, E Szabó, Hyperlinearity, essentially free actions and $L^2$–invariants. The sofic property, Math. Ann. 332 (2005) 421

[9] S Friedl, T Kim, T Kitayama, Poincaré duality and degrees of twisted Alexander polynomials, Indiana Univ. Math. J. 61 (2012) 147

[10] S L Harvey, Higher-order polynomial invariants of $3$–manifolds giving lower bounds for the Thurston norm, Topology 44 (2005) 895

[11] J Hempel, Residual finiteness for $3$–manifolds, from: "Combinatorial group theory and topology" (editors S M Gersten, J R Stallings), Ann. of Math. Stud. 111, Princeton Univ. Press (1987) 379

[12] W Li, W Zhang, An $L^2$–Alexander–Conway invariant for knots and the volume conjecture, from: "Differential geometry and physics" (editors M L Ge, W Zhang), Nankai Tracts Math. 10, World Sci. Publ. (2006) 303

[13] W Li, W Zhang, An $L^2$–Alexander invariant for knots, Commun. Contemp. Math. 8 (2006) 167

[14] W Lück, $L^2$–invariants: Theory and applications to geometry and $K$–theory, Ergeb. Math. Grenzgeb. 44, Springer (2002)

[15] J R Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company (1984)

[16] W P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 339, Amer. Math. Soc. (1986)

[17] V G Turaev, Euler structures, nonsingular vector fields, and Reidemeister-type torsions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 607, 672

[18] V G Turaev, Introduction to combinatorial torsions, Birkhäuser, Basel (2001)

[19] V G Turaev, Torsions of $3$–dimensional manifolds, Progress in Mathematics 208, Birkhäuser, Basel (2002)

[20] M Wada, Twisted Alexander polynomial for finitely presentable groups, Topology 33 (1994) 241

Cité par Sources :