Integrality of the Averaged Jones Polynomial of Algebraically Split Links

X.-S. Lin and Z. Wang recently made a conjecture concerning the integrality of the Taylor coefficients of the averaged Jones polynomial of algebraically split links. This question is related to a conjectural integrality result for the Ohtsuki invariants, which are rational topological invariants of homology 3-spheres derived from the quantum invariants of Witten and Reshetikhin-Turaev. This paper presents counterexamples to the conjecture of Lin and Wang and gives a proof of a corrected statement of the conjecture. The statement is divided into two parts. The first part applies to geometrically split links and gives an integrality result which is stronger than conjectured, while the second is valid for any algebraically split link and gives a somewhat weaker statement. Both propositions are seen to be sharp by the examples given.