Lattice conformation of theta-curves accompanied with Brunnian property

Abstract

A theta-curve is an embedding of the Greek letter Θ shaped graph in three-dimensional space. This is a useful physical model for polymer chains since theta-curve motifs are often present in many circular proteins with internal bridges. A Brunnian theta-curve is a nontrivial theta-curve with the property that if we remove any one among three edges, then the remaining knot can be laid in the plane without crossings. We focus on the rigidity of polymer chains with the Brunnian theta-curve shape by using the lattice stick number which is the minimal number of sticks glued end-to-end that are necessary to construct the theta-curve in the cubic lattice. The authors have already shown in a previous research that at least 15 lattice sticks are needed to construct Brunnian theta-curves. In this paper, we improve the lower bound of the lattice stick number for Brunnian theta-curves to 16.