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The ropelength is a mathematical quantity that regulates the tightness of flexible strands in the three-dimensional space. The superhelical conformation of long twisted strands is known to be more efficient in terms of ropelength compared with the circular double helical conformation. In this paper, we present a conformation of 2-bridge knots by using ropelength-minimizing superhelical curves and derive an upper bound on the ropelength of 2-bridge knots. Our superhelical model of 2-bridge knots is shown to be more efficient than the standard double helical one if the iterative twisted parts are long enough.

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Chirality is one of the important assymmetrical property in wide area of natural science, which has been studied to predict molecular behavior. One of good methods to analyze molecules with complex structures is representing them as graphs embedded in 3-dimensional space. So it is important to study the chirality of spatial graphs to understand structure of chiral molecules. Moreover, Robertson and Seymour's graph minor theorem implies that a set of minor minimal graphs with respect to intrinsic properties is finite. So it is also important to find a complete set of minor minimal graphs for intrinsic properties. In this paper, we classify minor minimal intrinsically chiral graphs among simple graphs of size up to twelve.

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 cross-ings. We are 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 by 16.