The architecture of polyhedral links and their HOMFLY polynomials

A general approach is proposed to elucidate the topological characteristics of molecules with the shape of polyhedral links. For an arbitrary polyhedral graph, four classes of polyhedral links can be obtained by applying the operation of ‘X-tangle covering’ to the related reduced sets. The relationships between the W-polynomial of a polyhedral graph and the HOMFLY polynomial of each kind of polyhedral links are established. These relationships not only simplify the computation but also provide a method of constructing a general formula for the HOMFLY polynomial of polyhedral links.     

The architecture of Platonic polyhedral links

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-branched 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.     

The braid index of polyhedral links

Polyhedral links have been used to model DNA polyhedra and protein catenanes. Some topological characteristics of a type of polyhedral links fabricated from a polyheron by the method of ‘n-branched curve and X-tangled covering’ have been elucidated. In this paper, we pay close attention to their braid index considered significant in view of DNA nanotechnology, and proved that the MFW inequality is sharp for the polyhedral links. Our results demonstrate that the braid index of the links is directed by their crossing numbers. In addition, the studies of the polyhedral links can facilitate the research of the properties of DNA molecules, and can characterize their structural complexity.     

Enumerating DNA polyhedral links

With the fast-growing of DNA self-assembly techniques, a large number of DNA catenanes or more specifically, DNA polyhedral links were synthesized in the past decade. As a sequel of a recent paper (J. Math. Chem. 50, p. 1693, 2012) by the present authors on the enumeration problem of polyhedral links, this paper considers the enumeration problem of DNA polyhedral links. In contrast to a general molecular link, a DNA polyhedral link has four notable features: 1. the memory of DNA chain direction; 2. the accurate DNA complementary base pairing; 3. the twist patterns of the double-helical strands; 4. the migration in branched junction. These features put forward particular requests for treating the enumeration problem of DNA polyhedral links. In addition to using the standard Pólya’s counting theory, we here introduce the generating function and edge direction retentivity analysis to the enumeration problem, by which we establish explicit expressions of the numbers of DNA polyhedral links for three typical models. These models have been used as strategies or are potential strategies in the synthesis of DNA polyhedral links.     

The complexity of Platonic and Archimedean polyhedral links

A mathematical methodology for understanding the complexity of Platonic and Archimedean polyhedral links has been developed based on some topological invariants from knot theory. Knot invariants discussed here include crossing number, unknotting number, genus and braid index, which are considered significant in view of DNA nanotechnology. Our results demonstrate that the braid index provides the most structural information; hence, it can be used, among four knot invariants, as the most useful complexity measure. Using such an invariant, it indicates that the complexity of polyhedral links is directed by the number of their building blocks. The research introduces a simple but important concept in the theoretical characterization and analysis of DNA polyhedral catenanes.     

The architecture and the Jones polynomial of polyhedral links

In this paper, we first recall some known architectures of polyhedral links (Qiu and Zhai in J Mol Struct (THEOCHEM) 756:163–166, 2005; Yang and Qiu in MATCH Commun Math Comput Chem 58:635–646, 2007; Qiu et al. in Sci China Ser B Chem 51:13–18, 2008; Hu et al. in J Math Chem 46:592–603, 2009; Cheng et al. in MATCH Commun Math Comput Chem 62:115–130, 2009; Cheng et al. in MATCH Commun Math Comput Chem 63:115–130, 2010; Liu et al. in J Math Chem 48:439–456 2010). Motivated by these architectures we introduce the notions of polyhedral links based on edge covering, vertex covering, and mixed edge and vertex covering, which include all polyhedral links in Qiu and Zhai (J Mol Struct (THEOCHEM) 756:163–166, 2005), Yang and Qiu (MATCH Commun Math Comput Chem 58:635–646, 2007), Qiu et al. (Sci China Ser B Chem 51:13–18, 2008), Hu et al. (J Math Chem 46:592–603, 2009), Cheng et al. (MATCH Commun Math Comput Chem 62:115–130, 2009), Cheng et al. (MATCH Commun Math Comput Chem 63:115–130, 2010), Liu et al. (J Math Chem 48:439–456, 2010) as special cases. The analysis of chirality of polyhedral links is very important in stereochemistry and the Jones polynomial is powerful in differentiating the chirality (Flapan in When topology meets chemistry. Cambridge Univ. Press, Cambridge, 2000). Then we give a detailed account of a result on the computation of the Jones polynomial of polyhedral links based on edge covering developed by Jin, Zhang, Dong and Tay (Electron. J. Comb. 17(1): R94, 2010) and, at the same time, by using this method we obtain some new computational results on polyhedral links of rational type and uniform polyhedral links with small edge covering units. These new computational results are helpful to judge the chirality of polyhedral links based on edge covering. Finally, we give some remarks and pose some problems for further study.     

Architecture of Platonic and Archimedean polyhedral links

A new methodology for understanding the construction of polyhedral links has been developed on the basis of the Platonic and Archimedean solids by using our method of the ‘three-cross-curve and dou- ble-twist-line covering’. There are five classes of polyhedral links that can be explored: the tetrahedral and truncated tetrahedral links; the hexahedral and truncated hexahedral links; the dodecahedral and truncated dodecahedral links; the truncated octahedral and icosahedral links. Our results show that the tetrahedral and truncated tetrahedral links have T symmetry; the hexahedral and truncated hexahedral links, as well as the truncated octahedral links, O symmetry; the dodecahedral and truncated dodeca- hedral links, as well as the truncated icosahedral links, I symmetry, respectively. This study provides further insight into the molecular design, as well as theoretical characterization, of the DNA and protein catenanes.     

Enumerating the total colorings of a polyhedron and application to polyhedral links

In this paper we consider the enumeration problem of a particular three-dimensional molecular or chemical compound system which has a polyhedral frame where the vertices, edges and faces represent ‘units’ such as atoms, bonds, ligands, polymers, or other objects of chemical interests. In this system, chirality is also taken into account. This enumeration problem is mathematically modeled as the ‘total coloring’ enumeration problem of a polyhedron: i.e., the number of ways to color all the vertices, edges and faces of the polyhedron by using three or more corresponding color sets, in which some colors may be chiral. We establish a general formula for this enumeration problem by extending the fundamental version of Pólya’s enumeration theorem. In particular, we apply this technique to the enumeration problem of polyhedral links which have received special attention from biochemists, mathematical chemists and mathematicians over the past two decades.     

Tutte and Jones polynomials of links,polyominoes and graphical recombination patterns

In order to facilitate recognition of polymer graphs and patterns obtained by graphical recombination, we analyze polynomial invariants, graphs and knots associated to polyominoes, polyiamonds, and polyhexes.     

The enumeration of electron-rich and electron-poor polyhedral clusters

We present as dual processes the capping of closed triangulated polyhedra with apical atoms and the making of holes in such polyhedra either by puncture of the surface or by excision of atoms and their edges. These processes are shown to generate stable chemical species containing respectively less or more than 2n + 2 skeletal electrons. The former species are designated as electron-poor whereas the latter are called electron-rich. Pólya's enumeration method is used to enumerate the distinct ways of capping and excising the closed, triangulated polyhedra to yield systems containing from four to twelve vertices. For the enumeration of cappings the appropriate cycle index is that of the dual of the polyhedron being capped, whilst for the enumeration of the excisions the cycle index is that of the polyhedron being excised.     

The mechanical properties of crystalline cyclopentyl polyhedral oligomeric silsesquioxane

The anisotropic elastic constants of crystalline octacyclopentyl polyhedral oligomeric silsesquioxane (CpPOSS) were determined using molecular dynamics. The force field used for these calculations was shown to model accurately the rhombohedral and triclinic crystal structures of octasilsesquioxane and CpPOSS, respectively, as well as the vibrational frequencies of octasilsesquioxane. The moduli for CpPOSS are anisotropic, with a Reuss-averaged bulk modulus of 7.5 GPa, an isotropic averaged Young's modulus of 11.78 GPa, and an isotropic averaged shear modulus of 4.75 GPa. These isotropic averages or, alternatively, the full anisotropic stiffness tensor of the crystal can be used with micromechanical composite models to calculate the effective elastic properties of polymer nanocomposites that contain crystalline aggregates of CpPOSS.     

Topology of polyhedral clathrate frameworks. 2. Frameworks constructed of polyhedral stories

The main features of the most widespread class of polyhedra clathrate frameworks formed of polyhedral stories are investigated. An algorithm for constructing such frameworks is suggested. Institute of Inorganic Chemistry, Siberian Branch, Russian Academy of Sciences. Translated fromZhurnal Struktumoi Khimii, Vol. 37, No. 1, pp. 123–130, January–February, 1996. Translated by L. Smolina     

The synthesis and characterisation of some closed polyhedral metallocarbon clusters

Ligands containing unsaturated C₂ and C₄ units have been reacted with triruthenium dodecacarbonyl to produce new organometallic clusters with simple closo-Ru_xC_y polyhedral frameworks which may be regarded as quasi-carboranes. The thermolysis of [Ru₃(CO)₁₂] with 1,4-diphenybutadiene yields the new clusters [Ru₃(CO)₈(μ₃-CPh(CH)₂CPh)] 2 and [Ru₄(CO)₉(μ₄-CPhCCH₂CH₂Ph)] 3, while treatmentof a solution of [Ru₃(CO)₁₂] and diphenylacetylene with trimethylamine N–oxide (Me₃NO) yields [Ru₂(CO)₆(μ-{C₂Ph₂}₂CO)] 4 as the major product and the new cluster [Ru₄(CO)₁₁(μ₄-C₂Ph₂)₂] 5. The solid-state structures of 2, 3 and 5 have been established by single crystal X-ray diffraction analyses and are shown to possess closo-Ru₃C₄ pentagonal bipyramidal, closo-Ru₄C₂ octahedral and closo-Ru₄C₄ dodecahedral skeletons, respectively. The structure and bonding in all three clusters may be rationalised using the Wade–Mingos polyhedral skeletal electron pair approach.     

F-structure of polyhedral water clusters

Energy and entropy characteristics of the F-structures of polyhedral water clusters containing tetragonal, pentagonal, and hexagonal faces with the intermolecular potential ST2 are studied.Institute of Thermal Physics, Ural Branch, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 34, No. 4, pp. 77–82, July–August, 1993.Translated by L. Smolina     

The Homfly polynomial of double crossover links

Motivated by double crossover DNA polyhedra (He et al. in Nature 452:198, 2008; Lin et al. in Biochemistry 48:1663, 2009; Zhang et al. in J Am Chem Soc 131:1413, 2009; Zhang et al. in Proc Natl Acad Sci USA 105:10665, 2008; He et al. in Angew Chem Int Ed 49:748, )2010, in this paper, we construct a new type of link, called the double crossover link, formed by utilizing the “ $n$ -point star” to cover each vertex of a connected graph $G$ . The double crossover link can be used to characterize the topological properties of double crossover DNA polyhedra. We show that the Homfly polynomial of the double crossover link can be obtained from the chain polynomial of the truncated graph of $G$ with two distinct labels. As an application, by using computer algebra (Maple) techniques, the Homfly polynomial of a double crossover tetrahedral link is obtained. Our result may be used to characterize and analyze the topological structure of DNA polyhedra.     

The formation of polyhedral ethylsilsesquioxanes in the process of oligoethylhydrocyclosiloxane polycondensation

A mechanism by which polyhedral oligoethylsilsesquioxanes are formed in the course of alkali-catalyzed polycondensation of oligoethylhydrocyclosiloxanes is suggested on the basis of the identification of 24 intermediates by GC-MS. The main trend in the formation of the silicon-oxygen framework of the oligoethylsilsesquioxanes is the consecutive addition of monomers, RSiHX₂ and dimers, (RSiHX)₂O, to the initial cyclosiloxanes and to the condensed ethylpolycyclosiloxanes thus formed. The mechanism suggested is confirmed by the data from a chromatographic investigation of the variations in the concentration of the reaction products.     

Supercrystal structures of polyhedral PbS nanocrystals

We analyzed the interaction between chemically grafted polysaccharide layers in aqueous solutions. To fabricate such layers, an end-terminated dextran silane coupling agent was synthesized and the polydextran was grafted to oxidized silicon wafers and to silica particles. This resulted in the formation of a 28 nm thick layer (in air) and a grafted amount of 40 mg/m(2) as determined by ellipsometry. The physical properties of the grafted layer were investigated in aqueous solutions by atomic force microscope imaging and colloidal probe force measurements. Surface and friction forces were measured between one bare and one polydextran coated silica surface. A notable feature was a bridging attraction due to affinity between dextran and the silica surface. Surface interactions and friction forces were also investigated between two surfaces coated with grafted polydextran. Repulsive forces were predominant, but nevertheless a high friction force was observed. The repulsive forces were enhanced by addition of sodium dodecyl sulfate (SDS) that associates with the tethered polydextran layers. SDS also decreased the friction force. Our data suggests that energy dissipation due to shear-induced structural changes within the grafted layer is of prime importance for the high friction forces observed, in particular deformation of protrusions in the surface layer.