A-polynomials, Ptolemy equations and Dehn filling

Howie, Joshua A¹ ; Mathews, Daniel V¹ ; Purcell, Jessica S¹

¹School of Mathematics, Monash University, Clayton VIC, Australia

Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1265-1320

Résumé

The A-polynomial encodes hyperbolic geometric information on knots and related manifolds. Historically, it has been difficult to compute, and particularly difficult to determine A-polynomials of infinite families of knots. Here, we compute A-polynomials by starting with a triangulation of a manifold, then using symplectic properties of the Neumann–Zagier matrix encoding the gluings to change the basis of the computation. The result is a simplification of the defining equations. We apply this method to families of manifolds obtained by Dehn filling, and show that the defining equations of their A-polynomials are Ptolemy equations which, up to signs, are equations between cluster variables in the cluster algebra of the cusp torus.

Détail

DOI : 10.2140/agt.2025.25.1265

Keywords: A-polynomial, gluing equations, triangulations, Ptolemy equations, Dehn filling, layered solid torus, Farey complex

Bibliographie

[1] C Adams, W Sherman, Minimum ideal triangulations of hyperbolic 3-manifolds, Discrete Comput. Geom. 6 (1991) 135 | DOI

[2] D W Boyd, Mahler’s measure and invariants of hyperbolic manifolds, from: "Number theory for the millennium, I", A K Peters (2002) 127

[3] B A Burton, R Budney, W Pettersson, Others, Regina: software for low-dimensional topology (1999–2019)

[4] A A Champanerkar, A-polynomial and Bloch invariants of hyperbolic 3-manifolds, PhD thesis, Columbia University (2003)

[5] D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of 3–manifolds, Invent. Math. 118 (1994) 47 | DOI

[6] D Cooper, D D Long, Remarks on the A-polynomial of a knot, J. Knot Theory Ramifications 5 (1996) 609 | DOI

[7] M Culler, A-polynomials, electronic resource (2013)

[8] M Culler, N M Dunfield, M Goerner, J R Weeks, SnapPy, a computer program for studying the geometry and topology of 3-manifolds (2016)

[9] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987) 237 | DOI

[10] M Culler, P B Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math. 117 (1983) 109 | DOI

[11] M Culler, P B Shalen, Bounded, separating, incompressible surfaces in knot manifolds, Invent. Math. 75 (1984) 537 | DOI

[12] T Dimofte, Quantum Riemann surfaces in Chern–Simons theory, Adv. Theor. Math. Phys. 17 (2013) 479 | DOI

[13] T Dimofte, R Van Der Veen, A spectral perspective on Neumann–Zagier, preprint (2014)

[14] V Fock, A Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006) 1 | DOI

[15] S Fomin, M Shapiro, D Thurston, Cluster algebras and triangulated surfaces, I : Cluster complexes, Acta Math. 201 (2008) 83 | DOI

[16] S Fomin, L Williams, A Zelevinsky, Introduction to cluster algebras : chapters 1–3, preprint (2016)

[17] C Frohman, R Gelca, W Lofaro, The A-polynomial from the noncommutative viewpoint, Trans. Amer. Math. Soc. 354 (2002) 735 | DOI

[18] S Garoufalidis, On the characteristic and deformation varieties of a knot, from: "Proceedings of the Casson Fest", Geom. Topol. Monogr. 7, Geom. Topol. Publ. (2004) 291 | DOI

[19] S Garoufalidis, T T Q Lê, The colored Jones function is q-holonomic, Geom. Topol. 9 (2005) 1253 | DOI

[20] S Garoufalidis, T W Mattman, The A-polynomial of the (−2,3,3 + 2n) pretzel knots, New York J. Math. 17 (2011) 269

[21] S Garoufalidis, D P Thurston, C K Zickert, The complex volume of SL(n, C)-representations of 3-manifolds, Duke Math. J. 164 (2015) 2099 | DOI

[22] M Gekhtman, M Shapiro, A Vainshtein, Cluster algebras and Weil–Petersson forms, Duke Math. J. 127 (2005) 291 | DOI

[23] M Goerner, C K Zickert, Triangulation independent Ptolemy varieties, Math. Z. 289 (2018) 663 | DOI

[24] F Guéritaud, S Schleimer, Canonical triangulations of Dehn fillings, Geom. Topol. 14 (2010) 193 | DOI

[25] J Y Ham, J Lee, An explicit formula for the A-polynomial of the knot with Conway’s notation C(2n,3), J. Knot Theory Ramifications 25 (2016) 1650057 | DOI

[26] K Hikami, R Inoue, Braids, complex volume and cluster algebras, Algebr. Geom. Topol. 15 (2015) 2175 | DOI

[27] J Hoste, P D Shanahan, A formula for the A-polynomial of twist knots, J. Knot Theory Ramifications 13 (2004) 193 | DOI

[28] J A Howie, D V Mathews, J S Purcell, E K Thompson, A-polynomials of fillings of the Whitehead sister, Int. J. Math. 34 (2023) 2350085 | DOI

[29] W Jaco, J H Rubinstein, 0-efficient triangulations of 3-manifolds, J. Differential Geom. 65 (2003) 61

[30] W Jaco, J H Rubinstein, Layered-triangulations of 3-manifolds, preprint (2006)

[31] D V Mathews, An explicit formula for the A-polynomial of twist knots, J. Knot Theory Ramifications 23 (2014) 1450044 | DOI

[32] W D Neumann, Combinatorics of triangulations and the Chern–Simons invariant for hyperbolic 3-manifolds, from: "Topology ’90", Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 243 | DOI

[33] W D Neumann, D Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) 307 | DOI

[34] M Newman, Integral matrices, 45, Academic Press (1972)

[35] Y Ni, X Zhang, Detection of knots and a cabling formula for A-polynomials, Algebr. Geom. Topol. 17 (2017) 65 | DOI

[36] R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299 | DOI

[37] K L Petersen, A-polynomials of a family of two-bridge knots, New York J. Math. 21 (2015) 847

[38] H Segerman, A generalisation of the deformation variety, Algebr. Geom. Topol. 12 (2012) 2179 | DOI

[39] P B Shalen, Representations of 3-manifold groups, from: "Handbook of geometric topology", North-Holland (2002) 955

[40] N Tamura, Y Yokota, A formula for the A-polynomials of (−2,3,1 + 2n)-pretzel knots, Tokyo J. Math. 27 (2004) 263 | DOI

[41] W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)

[42] W P Thurston, Three-dimensional geometry and topology, I, 35, Princeton Univ. Press (1997)

[43] A T Tran, The A-polynomial 2-tuple of twisted Whitehead links, Int. J. Math. 29 (2018) 1850013 | DOI

[44] J Weeks, Computation of hyperbolic structures in knot theory, from: "Handbook of knot theory", Elsevier (2005) 461 | DOI

[45] L K Williams, Cluster algebras: an introduction, Bull. Amer. Math. Soc. 51 (2014) 1 | DOI

[46] C K Zickert, Ptolemy coordinates, Dehn invariant and the A-polynomial, Math. Z. 283 (2016) 515 | DOI