Finiteness of the image of the Reidemeister torsion of a splice

Kitano, Teruaki; Nozaki, Yuta

doi:10.5802/ambp.389

Kitano, Teruaki ¹ ; Nozaki, Yuta ²

¹ Department of Information Systems Science, Faculty of Science and Engineering, Soka University Tangi-cho 1-236, Hachioji, Tokyo 192-8577, Japan
² Organization for the Strategic Coordination of Research and Intellectual Properties, Meiji University 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan

Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 1, pp. 19-36.

Résumé

The set $RT (M)$ of values of the $SL (2, ℂ)$ -Reidemeister torsion of a 3-manifold $M$ can be both finite and infinite. We prove that $RT (M)$ is a finite set if $M$ is the splice of two certain knots in the 3-sphere. The proof is based on an observation on the character varieties and $A$ -polynomials of knots.

Publié le : 2020-08-26

DOI : 10.5802/ambp.389

Classification : 57M27, 57M25, 20C99, 14M99
Mots clés : Reidemeister torsion, $A$-polynomial, character variety, splice, bending, Riley polynomial

Affiliations des auteurs :

Kitano, Teruaki ¹ ; Nozaki, Yuta ²

@article{AMBP_2020__27_1_19_0,
     author = {Kitano, Teruaki and Nozaki, Yuta},
     title = {Finiteness of the image of the {Reidemeister} torsion of a splice},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {19--36},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {27},
     number = {1},
     year = {2020},
     doi = {10.5802/ambp.389},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ambp.389/}
}

TY  - JOUR
AU  - Kitano, Teruaki
AU  - Nozaki, Yuta
TI  - Finiteness of the image of the Reidemeister torsion of a splice
JO  - Annales mathématiques Blaise Pascal
PY  - 2020
SP  - 19
EP  - 36
VL  - 27
IS  - 1
PB  - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
UR  - http://www.numdam.org/articles/10.5802/ambp.389/
DO  - 10.5802/ambp.389
LA  - en
ID  - AMBP_2020__27_1_19_0
ER  -

%0 Journal Article
%A Kitano, Teruaki
%A Nozaki, Yuta
%T Finiteness of the image of the Reidemeister torsion of a splice
%J Annales mathématiques Blaise Pascal
%D 2020
%P 19-36
%V 27
%N 1
%I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
%U http://www.numdam.org/articles/10.5802/ambp.389/
%R 10.5802/ambp.389
%G en
%F AMBP_2020__27_1_19_0

Kitano, Teruaki; Nozaki, Yuta. Finiteness of the image of the Reidemeister torsion of a splice. Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 1, pp. 19-36. doi : 10.5802/ambp.389. http://www.numdam.org/articles/10.5802/ambp.389/

Bibliographie
Cité par

[1] Abouzaid, Mohammed; Manolescu, Ciprian A sheaf-theoretic model for $SL (2, ℂ)$ Floer homology (to appear in J. Eur. Math. Soc.)

[2] Boden, Hans U.; Curtis, Cynthia L. Splicing and the ${SL}_{2} (ℂ)$ Casson invariant, Proc. Am. Math. Soc., Volume 136 (2008) no. 7, pp. 2615-2623 | DOI | MR

[3] Cooper, Daryl; Culler, Marc; Gillet, Henri; Long, Darren D.; Shalen, Peter B. Plane curves associated to character varieties of $3$ -manifolds, Invent. Math., Volume 118 (1994) no. 1, pp. 47-84 | DOI | MR | Zbl

[4] Cooper, Daryl; Long, Darren D. Remarks on the $A$ -polynomial of a knot, J. Knot Theory Ramifications, Volume 5 (1996) no. 5, pp. 609-628 | DOI | MR

[5] Curtis, Cynthia L. An intersection theory count of the ${SL}_{2} (ℂ)$ -representations of the fundamental group of a $3$ -manifold, Topology, Volume 40 (2001) no. 4, pp. 773-787 | DOI | MR

[6] Curtis, Cynthia L. Erratum to: “An intersection theory count of the ${SL}_{2} (ℂ)$ -representations of the fundamental group of a 3-manifold” [Topology 40(4):773–787, 2001], Topology, Volume 42 (2003) no. 4, 929 pages | DOI | MR

[7] Grünbaum, Branko Convex polytopes, Pure and Applied Mathematics, 16, Interscience Publishers, 1967, xiv+456 pages (With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard.) | MR

[8] Hatcher, Allen; Thurston, William Incompressible surfaces in $2$ -bridge knot complements, Invent. Math., Volume 79 (1985) no. 2, pp. 225-246 | DOI | MR

[9] Heusener, Michael ${SL}_{n} (ℂ)$ -representation spaces of knot groups, RIMS Kokyuroku, Volume 1991 (2016), pp. 1-26

[10] Johnson, Dennis A geometric form of Casson’s invariant and its connection to Reidemeister torsion (unpublished lecture notes)

[11] Johnson, Dennis; Millson, John J. Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis (New Haven, Conn., 1984) (Progress in Mathematics), Volume 67, Birkhäuser, 1987, pp. 48-106 | DOI | MR

[12] Kitano, Teruaki Reidemeister torsion of Seifert fibered spaces for $SL (2; C)$ -representations, Tokyo J. Math., Volume 17 (1994) no. 1, pp. 59-75 | DOI | MR

[13] Kitano, Teruaki Reidemeister torsion of the figure-eight knot exterior for $SL (2; C)$ -representations, Osaka J. Math., Volume 31 (1994) no. 3, pp. 523-532 | MR

[14] Milnor, John Two complexes which are homeomorphic but combinatorially distinct, Ann. Math., Volume 74 (1961), pp. 575-590 | DOI | MR

[15] Milnor, John A duality theorem for Reidemeister torsion, Ann. Math., Volume 76 (1962), pp. 137-147 | DOI | MR

[16] Morifuji, Takayuki Twisted Alexander polynomials of twist knots for nonabelian representations, Bull. Sci. Math., Volume 132 (2008) no. 5, pp. 439-453 | DOI | MR

[17] Motegi, Kimihiko Haken manifolds and representations of their fundamental groups in $SL (2, C)$ , Topology Appl., Volume 29 (1988) no. 3, pp. 207-212 | DOI | MR

[18] Paoluzzi, Luisa; Porti, Joan Non-standard components of the character variety for a family of Montesinos knots, Proc. Lond. Math. Soc., Volume 107 (2013) no. 3, pp. 655-679 | DOI | MR

[19] Riley, Robert Nonabelian representations of $2$ -bridge knot groups, Q. J. Math., Oxf. II. Ser., Volume 35 (1984) no. 138, pp. 191-208 | DOI | MR

[20] Zentner, Raphael Integer homology 3-spheres admit irreducible representations in $SL (2, ℂ)$ , Duke Math. J., Volume 167 (2018) no. 9, pp. 1643-1712 | DOI | MR

Cité par Sources :