A new methodology for understanding the construction of polyhedral links has been developed on the basis of the Platonic solids by using our method of the 'n-branched curves and m-twisted double-lines covering'. There are five classes of platonic polyhedral links we can construct: the tetrahedral links; the hexahedral links; the octahedral links; the dodecahedral links; the icosahedral links. The tetrahedral links, hexahedral links, and dodecahedral links are, respectively, assembled by using the method of the '3-branched curves and m-twisted double-lines covering', whereas the octahedral links and dodecahedral links are, respectively, made by using the method of the '4-ranched curves' and '5-branched curves', as well as 'm-twisted double-lines covering'. Moreover, the analysis relating topological properties and link invariants is of considerable importance. Link invariants are powerful tools to classify and measure the complexity of polyhedral catenanes. This study provides further insight into the molecular design, as well as theoretical characterization, of the DNA polyhedral catenanes.

Polyhedral links, interlinked and interlocked architectures, have been proposed for the description and analysis of knotted configurations in DNA and proteins. Qiu et al. fabricated cubic polyhedral and carbon nanotube links by the means of three cross curves and double lines covering, and analyzed their chirality by point groups. We present, in this paper, a novel method by replacing three cross curves with branched alternating closed braids to construct a new type of polyhedral links on arbitrary convex polyhedra. We give some conditions to determine the chirality of the polyhedral links in terms of generalized Tutte and Kauffman polynomials. Accordingly, we show that each regular branched closed braid link is chiral. This result shows that the model of bacteriophage HK97 capsid, topologically linked protein catenane, is chiral.

Current synthetic biology has witnessed a revolution that natural DNA molecule steps onto a broad scientific area by assembling a large variety of threedimensional structures with the connectivity of polyhedra. A mathematical model of these biomolecules is crucial to clarify the biological self-assembly principle, and unravel a first-step understanding of biological regulation and controlling mechanisms. In this paper, mechanisms of two different enzymatic actions on DNA polyhedra are elucidated through theoretical models of polyhedral links: (1) topoisomerase that untangles DNA polyhedral links produces separated single-stranded DNA circles through the crossing change operation; (2) recombinase generates a class of polyhedral circular paths or polyhedral knots by applying the crossing smoothing operation. Furthermore, we also discuss the possibility of applying two theoretical operations in molecular design of DNA polyhedra. Thus, our research provides a new sight of how geometry and topology of DNA polyhedra can be manipulated and controlled by enzymes, as well as has implications for molecular design and structural analysis of structural genome organization.

A new method for understanding the relations between molecular design and topological features has been developed on the basis of the Seifert construction in knot theory. Our result shows that the T2-molecular doubled knots possess the point symmetry group C2 and that the T2k-molecular doubled knots possess the point symmetry group C1 Hence both sets are topological chiral. When the rungs are cut down, topological symmetries of the molecular knots are unchanged, Our results led us to infer that the point symmetry group S1 is necessary and sufficient for the molecular topological rubber glove.

A new method for understanding the relations between the molecular design and topological chirality has been developed on the basis of the Seifert construction of knot theory. Tis interesting problem of topological chirality has been solved by applying the topological symmetry concept to the molecular mobius ladders with citber 2n+1 or 2n half-twists (n≥1). Our results show that the T2n+1-Mobius ladders possess the point symmetry group C2n+1 when Lγ=(2n+1)m. and that the Tπ-Mobius ladders possess the point symmetry group Cπ when Lγ=2nm or Lγ=2nm+n(m=1,2,…). Hence both sets are topological chiral.