Polygons with Prescribed Turning Angles in Three Dimensions: A Counterexample to a Conjecture of Efrat, Fulek, Kobourov and Tóth, and the Corrected Criterion

A closed polygon in \(\mathbb{R}^3\) has at each vertex a turning angle in \((0,\pi)\), and a sequence \(A=(\alpha_0,\dots,\alpha_{n-1})\in(0,\pi)^n\) is called realizable if it is the sequence of turning angles of such a polygon. Efrat, Fulek, Kobourov and Tóth conjectured (Oberwolfach Rep. 17 (2020); J. Graph Algorithms Appl. 26 (2022)) that a realizable sequence has a realization without self-intersections if and only if \(n\) is even or \(\sum_i(\pi-\alpha_i)\neq\pi\), and they proved the ‘only if’ part. We show that the conjecture holds for \(n=4\) and \(n=5\) and fails for every \(n\ge6\). For \(n=6\) the sequence \((\tfrac{5\pi}{6},\tfrac{5\pi}{6},\tfrac{5\pi}{6},\tfrac{5\pi}{6},\tfrac{5\pi}{6},\tfrac{\pi}{6})\) is realized by an explicit planar hexagon, while every polygon with these turning angles lies in a plane and has total signed turning \(\pm4\pi\), so that none is simple. The corrected criterion is the following. For a set \(S\) of indices put \(e_S(A)=\sum_{i\in S}(\pi-\alpha_i)+\sum_{i\notin S}\alpha_i-\pi\). A realizable sequence has a realization without self-intersections if and only if \(e_S(A)\ne0\) for every set \(S\) of odd cardinality at least \(5\); the conjecture accounts only for the set of all indices. If moreover \(\sum_i\alpha_i>2\pi\) and \(e_S(A)\neq0\) for all sets \(S\) of odd cardinality, there is such a realization which is not contained in a plane. The ingredients on the sphere are known and are not claimed as new: the inequalities \(e_S(A)\ge0\) for closed spherical polygons with prescribed side lengths, and their equality case (all vertices on a great circle), belong to the theory of spherical polygonal linkages (Galitzer, Biswas, Kapovich and Millson, Buckman and Schmitt, Mondello and Panov); the realizability criterion and the ‘only if’ part are due to the authors of the conjecture, who also assert, without proof, that crossings can be avoided whenever some realization is not contained in a plane. What this note adds is the observation that the equality case for a set of at least five indices other than the set of all indices is again an obstruction to simple realizations, the counterexamples, a proof of the assertion just mentioned (by an induction which inserts one vertex at a time), and the resulting characterization. The explicit hexagon is verified in exact arithmetic; all other computations are tests and are not used in the proofs. This is an unrefereed note.