Narrow Results

We give a self-contained treatment of Le and Habiro's approach to the Jones function of a knot and Habiro's cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a knot. As a corollary, we obtain bounds on the growth of coefficients in the Ohtsuki series for manifolds obtained by surgery around a knot, which support the slope conjecture of Jacoby and the first author.

We calculate the Ohtsuki invariants ${\lambda }_i(M)$$(i=0,1,2,3)$ of every Brieskorn–Hamm manifold $M$ which is a rational homology $3$-sphere. We denote the order of $H_1(M;\mathbb{Z})$ by $H$. By the result, we show that the third Ohtsuki invariant ${\lambda }_3(M)$ of the Brieskorn–Hamm manifolds $M$ with $H=1$ is negative, and the third Ohtsuki invariant ${\lambda }_3(M)$ of most Brieskorn–Hamm manifolds $M$ with $H\ge 2$ is positive.