The Complexity of Finding Tangles

We study the following combinatorial problem. Given a set of $n$ y-monotone curves, which we call wires, a tangle determines the order of the wires on a number of horizontal layers such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset $L$ of swaps (that is, unordered pairs of wires) and an initial order of the wires, a tangle realizes $L$ if each pair of wires changes its order exactly as many times as specified by $L$. Finding a tangle that realizes a given multiset of swaps and uses the least number of layers is known to be NP-hard. We show that it is even NP-hard to decide if a realizing tangle exists