  In this paper we study the relation between the function $J_{4_1,0}$, which
arises from a quantum invariant of the figure-eight knot, and Sudler's
trigonometric product. We find $J_{4_1,0}$ up to a constant factor along
continued fraction convergents to a quadratic irrational, and we show that its
asymptotics deviates from the universal limiting behavior that has been found
by Bettin and Drappeau in the case of large partial quotients. We relate the
value of $J_{4_1,0}$ to that of Sudler's trigonometric product, and establish
asymptotic upper and lower bounds for such Sudler products in response to a
question of Lubinsky.
