  In this paper, we study knot diagrams for which the underlying graph has
treewidth two. We give a linear time algorithm for the following problem: given
a knot diagram of treewidth two, does it represent the unknot? We also show
that for a link diagram of treewidth two we can test in linear time if it
represents the unlink. From the algorithm, it follows that a diagram of the
unknot of treewidth 2 can always be reduced to the trivial diagram with at most
$n$ (un)twist and (un)poke Reidemeister moves.
