ABSTRACT
The fact that these are nontrivial knots provides a new proof that Kinoshita's theta-curve is nonplanar, and furthermore, the fact that these knots are chiral shows that Kinoshita's theta-curve is chiral. The chapter shows that the planarity of a locally unlinked theta-curve or handcuff graph is entirely dependent on the planarity of an associated link. Let G be any theta-curve, or any handcuff graph whose consituent link has linking number zero. Let T(G) be the set of normalized tangles that can be used to describe G. The associated links and knots may not detect the nonplanarity of theta-curves and handcuff graphs in the case where the graphs themselves contain nontrivial knots or links. In the proof allowed for the possibility that h could induce any of twelve possible mappings of G to itself, that is, permuting the three arcs and exchanging the two vertices.
