  How should we thread a single string through a set of tubes so that pulling
the string taut self-assembles the tubes into a desired graph? While prior work
[ITCS 2024] solves this problem with the goal of minimizing the length of
string, we study here the objective of minimizing the total turn cost. The
frictional force required to pull the string through the tubes grows
exponentially with the total absolute turn angles (by the Capstan equation), so
this metric often dominates the friction in real-world applications such as
deployable structures. We show that minimum-turn threading is NP-hard, even for
graphs of maximum degree 4, and even when restricted to some special cases of
threading. On the other hand, we show that these special cases can in fact be
solved efficiently for graphs of maximum degree 4, thereby fully characterizing
their dependence on maximum degree. We further provide polynomial-time exact
and approximation algorithms for variants of turn-cost threading: restricting
to threading each edge exactly twice, and on rectangular grid graphs.
