On Habiro's Cyclotomic Expansions of the Ohtsuki Invariant

Abstract

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.

A Formula for the Colored Jones Polynomial of 2-Bridge Knots

“…Using Corollary 5, we give explicit formulas for the trefoil knot S(3, 1) and the figure eight knot S (5,3). By the formula (15) with p 1 = 1 and p = 1, we get…”

“…In this paper, we give a formula for the colored Jones polynomial of the 2-bridge knot S(p, t), using a formula obtained by R. Lawrence and O. Ron in [5]. As an application of our formula, we obtain a relation between the degree of the colored Jones polynomial and the crossing number for the 2-bridge knots S(4p + 1, 2p + 1) and S(4p − 1, 2p − 1).…”

Untitled

“…Furthermore the expression (3.13) exactly coincides with the form given by Le in Refs. 30, 31, where the WRT invariant was computed by use of Habiro's cyclotomic expansion of the N-colored Jones polynomial for trefoil [13,14,28,33] …”