Twisted Neumann-Zagier matrices. (English) Zbl 1533.81027
Summary: The Neumann-Zagier matrices of an ideal triangulation are integer matrices with symplectic properties whose entries encode the number of tetrahedra that wind around each edge of the triangulation. They can be used as input data for the construction of a number of quantum invariants that include the loop invariants, the 3D-index and state-integrals. We define a twisted version of Neumann-Zagier matrices, describe their symplectic properties, and show how to compute them from the combinatorics of an ideal triangulation. As a sample application, we use them to define a twisted version of the 1-loop invariant (a topological invariant) which determines the 1-loop invariant of the cyclic covers of a hyperbolic knot complement, and conjecturally is equal to the adjoint twisted Alexander polynomial.
MSC:
| 81P68 | Quantum computation |
| 16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |
| 20N05 | Loops, quasigroups |
| 05B40 | Combinatorial aspects of packing and covering |
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
| 32B25 | Triangulation and topological properties of semi-analytic and subanalytic sets, and related questions |
| 57K10 | Knot theory |
| 32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |
| 15A04 | Linear transformations, semilinear transformations |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
Keywords:
torsion; 1-loop invariant; adjoint Reidemeister torsion; Infinite cyclic cover; twisted Alexander polynomial; ideal triangulations; knots; hyperbolic 3-manifolds; Neumann-Zagier matrices; twisted Neumann-Zagier matrices; block circulant matricesSoftware:
SnapPyReferences:
| [1] | Andersen, J.E., Garoufalidis, S., Kashaev, R.: The volume conjecture for the klv state-integral, Preprint (2021) |
| [2] | Culler, M., Dunfield, N., Weeks, J.: SnapPy, a computer program for studying the topology of \(3\)-manifolds, Available at http://snappy.computop.org |
| [3] | Choi, Y-E, Neumann and Zagier’s symplectic relations, Expo. Math., 24, 1, 39-51 (2006) · Zbl 1136.57300 · doi:10.1016/j.exmath.2005.06.001 |
| [4] | Davis, P., Circulant Matrices (1979), New York: Wiley, New York · Zbl 0418.15017 |
| [5] | Dunfield, N.; Friedl, S.; Jackson, N., Twisted Alexander polynomials of hyperbolic knots, Exp. Math., 21, 4, 329-352 (2012) · Zbl 1266.57008 · doi:10.1080/10586458.2012.669268 |
| [6] | Dimofte, T.; Garoufalidis, S., The quantum content of the gluing equations, Geom. Topol., 17, 3, 1253-1315 (2013) · Zbl 1283.57017 · doi:10.2140/gt.2013.17.1253 |
| [7] | Dubois, J.; Garoufalidis, S., Rationality of the \({{\rm SL}}(2,{\mathbb{C} })\)-Reidemeister torsion in dimension 3, Topol. Proc., 47, 115-134 (2016) · Zbl 1333.57034 |
| [8] | Dimofte, T.; Garoufalidis, S., Quantum modularity and complex Chern-Simons theory, Commun. Number Theory Phys., 12, 1, 1-52 (2018) · Zbl 1447.57014 · doi:10.4310/CNTP.2018.v12.n1.a1 |
| [9] | Dimofte, T.; Gaiotto, D.; Gukov, S., 3-manifolds and 3d indices, Adv. Theor. Math. Phys., 17, 5, 975-1076 (2013) · Zbl 1297.81149 · doi:10.4310/ATMP.2013.v17.n5.a3 |
| [10] | Dimofte, T.; Gaiotto, D.; Gukov, S., Gauge theories labelled by three-manifolds, Commun. Math. Phys., 325, 2, 367-419 (2014) · Zbl 1292.57012 · doi:10.1007/s00220-013-1863-2 |
| [11] | Dubois, J.; Yamaguchi, Y., The twisted Alexander polynomial for finite abelian covers over three manifolds with boundary, Algebr. Geom. Topol., 12, 2, 791-804 (2012) · Zbl 1270.57020 · doi:10.2140/agt.2012.12.791 |
| [12] | Fox, R., Free differential calculus. III. Subgroups, Ann. Math. (2), 64, 407-419 (1956) · Zbl 0073.25401 · doi:10.2307/1969592 |
| [13] | Garoufalidis, S., Kashaev, R.: A meromorphic extension of the 3D index. Res. Math. Sci. 6(1), Paper No. 8, 34 (2019) · Zbl 1434.57020 |
| [14] | Garoufalidis, S., Storzer, M., Wheeler, C.: Perturbative invariants of cusped hyperbolic 3-manifolds, Preprint (2021), arXiv:2305.14884 |
| [15] | Garoufalidis, S., Yoon, S.: 1-loop equals torsion for fibered 3-manifolds, Preprint (2023), arXiv:2304.00469 |
| [16] | Garoufalidis, S., Yoon, S.: Asymptotically multiplicative quantum invariants, Preprint, (2022), arXiv:2211.00270 |
| [17] | Garoufalidis, S., Yoon, S.: Super-representations of 3-manifolds and torsion polynomials, Preprint (2023), arXiv:2301.11018 |
| [18] | Kitano, T., Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math., 174, 2, 431-442 (1996) · Zbl 0863.57001 · doi:10.2140/pjm.1996.174.431 |
| [19] | Kirk, P.; Livingston, C., Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology, 38, 3, 635-661 (1999) · Zbl 0928.57005 · doi:10.1016/S0040-9383(98)00039-1 |
| [20] | Kashaev, R.; Luo, F.; Vartanov, G., A TQFT of Turaev-Viro type on shaped triangulations, Ann. Henri Poincaré, 17, 5, 1109-1143 (2016) · Zbl 1337.81105 · doi:10.1007/s00023-015-0427-8 |
| [21] | Neumann, W.: Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic \(3\)-manifolds, Topology ’90 (Columbus, OH, 1990), Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, (1992), pp. 243-271 · Zbl 0768.57006 |
| [22] | Neumann, W.; Zagier, D., Volumes of hyperbolic three-manifolds, Topology, 24, 3, 307-332 (1985) · Zbl 0589.57015 · doi:10.1016/0040-9383(85)90004-7 |
| [23] | Porti, J.: Torsion de Reidemeister pour les variétés hyperboliques. Mem. Amer. Math. Soc. 128(612), x+139 (1997) · Zbl 0881.57020 |
| [24] | Ruiz-Claeyssen, JC; Leal, LAS, Diagonalization and spectral decomposition of factor block circulant matrices, Linear Algebra Appl., 99, 41-61 (1988) · Zbl 0641.15010 · doi:10.1016/0024-3795(88)90124-3 |
| [25] | Siejakowski, R.: Infinitesimal gluing equations and the adjoint hyperbolic Reidemeister torsion, arXiv:1710.02109, Preprint (2017) · Zbl 1497.57028 |
| [26] | Thurston, W.: The geometry and topology of 3-manifolds, Universitext, Springer-Verlag, Berlin, (1977), http://msri.org/publications/books/gt3m |
| [27] | Wada, M., Twisted Alexander polynomial for finitely presentable groups, Topology, 33, 2, 241-256 (1994) · Zbl 0822.57006 · doi:10.1016/0040-9383(94)90013-2 |
| [28] | Yamaguchi, Y., A relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion, Ann. Inst. Fourier (Grenoble), 58, 1, 337-362 (2008) · Zbl 1158.57027 · doi:10.5802/aif.2352 |
| [29] | Zickert, C., Ptolemy coordinates, Dehn invariant and the \(A\)-polynomial, Math. Z., 283, 1-2, 515-537 (2016) · Zbl 1354.57027 · doi:10.1007/s00209-015-1608-3 |
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