Knot theory and cluster algebra III: Posets

In previous work, we associated a module $T(i)$ to every segment $i$ of a link diagram $K$ and showed that there is a poset isomorphism between the submodules of $T(i)$ and the Kauffman states of $K$ relative to $i$. In this paper, we show that the posets are distributive lattices and give explicit descriptions of the join irreducibles in both posets. We also prove that the subposet of join irreducible Kauffman states is isomorphic to the poset of the coefficient quiver of $T(i)$.