Topology-based optimization of handcuff graphs on 3D lattice

In the field of macromolecular chemistry, handcuff-shaped catenanes and pretzelanes have a conformation consisting of two distinct loops and an edge connecting them. In spatial graph theory, this shape is referred to as a handcuff graph. One topological aspect of interest in these molecular structures involves determining the minimal number of monomers required to create them. In this paper, we focus on a handcuff graph situated in the cubic lattice, which we refer to as a lattice handcuff graph. We explicitly verify that constructing a lattice handcuff graph requires at least 14 lattice sticks, except for the two handcuff graphs: the trivial handcuff graph and the Hopf-linked handcuff graph. Mainly we employ the properly leveled lattice conformation argument, which was developed by the authors to find the lattice stick number of knot-shaped and link-shaped molecules.