Knots, perturbative series and quantum modularity.(English)Zbl 07897506
Summary: We introduce an invariant of a hyperbolic knot which is a map \(\alpha\mapsto\mathbf{\Phi}_\alpha(h)\) from \(\mathbb{Q}/\mathbb{Z}\) to matrices with entries in \(\overline{\mathbb{Q}}[[h]]\) and with rows and columns indexed by the boundary parabolic \(\mathrm{SL}_2(\mathbb{C})\) representations of the fundamental group of the knot. These matrix invariants have a rich structure: (a) their \((\sigma_0,\sigma_1)\) entry, where \(\sigma_0\) is the trivial and \(\sigma_1\) the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity \(\mathrm{e}^{2\pi\mathrm{i}\alpha}\) as an element of the Habiro ring, and the remaining entries belong to generalized Habiro rings of number fields; (b) the first column is given by the perturbative power series of Dimofte-Garoufalidis; (c) the columns of \(\mathbf{\Phi}\) are fundamental solutions of a linear \(q\)-difference equation; (d) the matrix defines an \(\mathrm{SL}_2(\mathbb{Z})\)-cocycle \(W_\gamma\) in matrix-valued functions on \(\mathbb{Q}\) that conjecturally extends to a smooth function on \(\mathbb{R}\) and even to holomorphic functions on suitable complex cut planes, lifting the factorially divergent series \(\mathbf{\Phi}(h)\) to actual functions. The two invariants \(\mathbf{\Phi}\) and \(W_\gamma\) are related by a refined quantum modularity conjecture which we illustrate in detail for the three simplest hyperbolic knots, the \(4_1, 5_2\) and \((-2,3,7)\) pretzel knots. This paper has two sequels, one giving a different realization of our invariant as a matrix of convergent \(q\)-series with integer coefficients and the other studying its Habiro-like arithmetic properties in more depth.
MSC:
| 57K30 | General topology of 3-manifolds |
| 57K16 | Finite-type and quantum invariants, topological quantum field theories (TQFT) |
| 57K14 | Knot polynomials |
| 57K10 | Knot theory |
Keywords:
quantum topology;knots;3-manifolds;Jones polynomial;Kashaev invariant;volume conjecture;Chern-Simons theory;asymptotics;quantum modularity conjecture;quantum modular forms;hyperbolic 3-manifolds;dilogarithm;cocycles;\(\mathrm{SL}_2(\mathbb{Z})\);denominators;Habiro-like functions;functions near \(\mathbb{Q}\);Neumann-Zagier matrices;Nahm sums;\(q\)-holonomic modulesCite
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