Vassiliev invariants and a strange identity related to the Dedekind eta-function.

*(English)*Zbl 0989.57009The author studies the series \(F(q)=\sum_{n=0}^\infty(1-q)(1-q^2)\cdots(1-q^n)\), which terminates at each root of unity, but diverges when \(|q|\neq 1\). Thus \(F(q)\) may be thought of as a function on the roots of unity. Also, \(F(q)\) makes sense as a formal power series in \(\zeta-q\) for any root of unity \(\zeta\).

For \(D\geq 0\) let \(\xi_D\) denote the number of “regular linearized chord diagrams” of degree \(D\), introduced by A. Stoimenow [J. Knot Theory Ramifications 7, No. 1, 93-114 (1998; Zbl 0892.57003)]. It is shown that the \(D\)th coefficient in the expansion of \(F(q)\) in \(1-q\) is equal to \(\xi_D\). This yields an asymptotic formula of the form \(\xi_D\sim D!\sqrt{D}(\pi^2/6)^{-D}(C_0+C_1D^{-1}+C_2D^{-2}+\cdots)\) with computable constants \(C_i\), which implies that the dimension of the space of Vassiliev invariants of classical knots of degree \(D\) modulo those of degree \(<D\) is \(O(D!\sqrt{D}(\pi^2/6)^{-D})\). The author also proves a formula for the number of “primitive regular linearized chord diagrams”, which yields an upper bound for the dimensions of the spaces of primitive Vassiliev invariants.

It is also observed that, for each root of unity \(\zeta\), \(F(\zeta)\) is equal to the radial limit at \(q=\zeta\) of the function \(-(1/2)\sum_{n=1}^\infty n\chi(n)q^{(n^2-1)/24} =-(1/2)1-5q-7q^2+11q^5+13q^7-\cdots\) defined for \(|q|<1\), where \(\chi=(12/\cdot)\) is the unique Dirichlet character of conductor \(12\), which may be thought of as the “derivative of order one-half” of the Dedekind eta-function.

The expansion of \(e^{-t/24}F(e^{-t})\) in \(t\) and similar expansions at other roots of unity is studied. Also, a kind of modular property of the function \(\varphi:{\mathbb Q}\to{\mathbb C}\), \(\varphi(\alpha)=e^{\pi i\alpha/12} F(e^{2\pi i\alpha})\), is obtained.

Functions similar to the ones that appear in this paper also appear in R. Lawrence and D. Zagier [Asian J. Math. 3, No. 1, 93-107 (1999; Zbl 1024.11028)] in the context of the Witten-Reshetikhin-Turaev invariant of \(3\)-manifolds.

For \(D\geq 0\) let \(\xi_D\) denote the number of “regular linearized chord diagrams” of degree \(D\), introduced by A. Stoimenow [J. Knot Theory Ramifications 7, No. 1, 93-114 (1998; Zbl 0892.57003)]. It is shown that the \(D\)th coefficient in the expansion of \(F(q)\) in \(1-q\) is equal to \(\xi_D\). This yields an asymptotic formula of the form \(\xi_D\sim D!\sqrt{D}(\pi^2/6)^{-D}(C_0+C_1D^{-1}+C_2D^{-2}+\cdots)\) with computable constants \(C_i\), which implies that the dimension of the space of Vassiliev invariants of classical knots of degree \(D\) modulo those of degree \(<D\) is \(O(D!\sqrt{D}(\pi^2/6)^{-D})\). The author also proves a formula for the number of “primitive regular linearized chord diagrams”, which yields an upper bound for the dimensions of the spaces of primitive Vassiliev invariants.

It is also observed that, for each root of unity \(\zeta\), \(F(\zeta)\) is equal to the radial limit at \(q=\zeta\) of the function \(-(1/2)\sum_{n=1}^\infty n\chi(n)q^{(n^2-1)/24} =-(1/2)1-5q-7q^2+11q^5+13q^7-\cdots\) defined for \(|q|<1\), where \(\chi=(12/\cdot)\) is the unique Dirichlet character of conductor \(12\), which may be thought of as the “derivative of order one-half” of the Dedekind eta-function.

The expansion of \(e^{-t/24}F(e^{-t})\) in \(t\) and similar expansions at other roots of unity is studied. Also, a kind of modular property of the function \(\varphi:{\mathbb Q}\to{\mathbb C}\), \(\varphi(\alpha)=e^{\pi i\alpha/12} F(e^{2\pi i\alpha})\), is obtained.

Functions similar to the ones that appear in this paper also appear in R. Lawrence and D. Zagier [Asian J. Math. 3, No. 1, 93-107 (1999; Zbl 1024.11028)] in the context of the Witten-Reshetikhin-Turaev invariant of \(3\)-manifolds.

Reviewer: Kazuo Habiro (Kyoto)

##### MSC:

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

11F20 | Dedekind eta function, Dedekind sums |

05A30 | \(q\)-calculus and related topics |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11F11 | Holomorphic modular forms of integral weight |