A Reappearence of Wheels

Recently, a number of authors [KS, Oh2, Ro] have independently shown that the universal finite type invariant of rational homology 3-spheres on the level of sl2 can be recovered from the Reshetikhin-Turaev sl2 invariant. An important role in Ohtsuki’s proof [Oh3] plays a map j1 (which joins the legs of 2-legged chinese characters) and its relation to a map α in terms of a power series (on the level of sl2). The purpose of the present note is to give a universal formula of the map α in terms of a power series F of wheel chinese characters. The above formula is similar to universal formulas of wheel chinese characters considered in [BGRT1] and leads to a simple conceptual proof of the above mentioned relation between the maps j1 and α on the level of sl2.