Finiteness of the image of the Reidemeister torsion of a splice
Annales Mathématiques Blaise Pascal, Tome 27 (2020) no. 1, pp. 19-36.

The set RT(M) of values of the SL(2,)-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 :
DOI : https://doi.org/10.5802/ambp.389
Classification : 57M27,  57M25,  20C99,  14M99
Mots clés : Reidemeister torsion, A-polynomial, character variety, splice, bending, Riley polynomial
@article{AMBP_2020__27_1_19_0,
     author = {Teruaki Kitano and Yuta Nozaki},
     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 = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.389/}
}
Teruaki Kitano; Yuta Nozaki. 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. https://ambp.centre-mersenne.org/articles/10.5802/ambp.389/

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

[2] Hans U. Boden; Cynthia L. Curtis Splicing and the SL 2 () Casson invariant, Proc. Am. Math. Soc., Volume 136 (2008) no. 7, pp. 2615-2623 | Article | MR 2390534

[3] Daryl Cooper; Marc Culler; Henri Gillet; Darren D. Long; Peter B. Shalen Plane curves associated to character varieties of 3-manifolds, Invent. Math., Volume 118 (1994) no. 1, pp. 47-84 | Article | MR 1288467 | Zbl 0842.57013

[4] Daryl Cooper; Darren D. Long Remarks on the A-polynomial of a knot, J. Knot Theory Ramifications, Volume 5 (1996) no. 5, pp. 609-628 | Article | MR 1414090

[5] Cynthia L. Curtis An intersection theory count of the SL 2 ()-representations of the fundamental group of a 3-manifold, Topology, Volume 40 (2001) no. 4, pp. 773-787 | Article | MR 1851563

[6] Cynthia L. Curtis Erratum to: “An intersection theory count of the SL 2 ()-representations of the fundamental group of a 3-manifold” [Topology 40(4):773–787, 2001], Topology, Volume 42 (2003) no. 4, 929 pages | Article | MR 1958534

[7] Branko Grünbaum Convex polytopes, Pure and Applied Mathematics, Volume 16, Interscience Publishers, 1967, xiv+456 pages (With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard.) | MR 0226496

[8] Allen Hatcher; William Thurston Incompressible surfaces in 2-bridge knot complements, Invent. Math., Volume 79 (1985) no. 2, pp. 225-246 | Article | MR 778125

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

[10] Dennis Johnson A geometric form of Casson’s invariant and its connection to Reidemeister torsion (unpublished lecture notes)

[11] Dennis Johnson; John J. Millson 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 | Article | MR 900823

[12] Teruaki Kitano Reidemeister torsion of Seifert fibered spaces for SL (2;C)-representations, Tokyo J. Math., Volume 17 (1994) no. 1, pp. 59-75 | Article | MR 1279569

[13] Teruaki Kitano 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 1309401

[14] John Milnor Two complexes which are homeomorphic but combinatorially distinct, Ann. Math., Volume 74 (1961), pp. 575-590 | Article | MR 0133127

[15] John Milnor A duality theorem for Reidemeister torsion, Ann. Math., Volume 76 (1962), pp. 137-147 | Article | MR 0141115

[16] Takayuki Morifuji Twisted Alexander polynomials of twist knots for nonabelian representations, Bull. Sci. Math., Volume 132 (2008) no. 5, pp. 439-453 | Article | MR 2426646

[17] Kimihiko Motegi Haken manifolds and representations of their fundamental groups in SL (2,C), Topology Appl., Volume 29 (1988) no. 3, pp. 207-212 | Article | MR 953952

[18] Luisa Paoluzzi; Joan Porti 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 | Article | MR 3100780

[19] Robert Riley Nonabelian representations of 2-bridge knot groups, Q. J. Math., Oxf. II. Ser., Volume 35 (1984) no. 138, pp. 191-208 | Article | MR 745421

[20] Raphael Zentner Integer homology 3-spheres admit irreducible representations in SL (2,), Duke Math. J., Volume 167 (2018) no. 9, pp. 1643-1712 | Article | MR 3813594