Abstract
In this paper, the authors study the mathematical properties of a class of alternating links called polyhedral links which have been used to model DNA polyhedra. The motivation of such studies is to provide guidance and aid in the research of the properties of certain DNA molecules. For example, such studies can provide characterizations of the structural complexity of DNA molecules. In an earlier work, Cheng and Jin studied the mathematical properties of such polyhedral links and were able to determine the braid index of a double crossover polyhedral link with 4 turn. However, the braid index of a double crossover polyhedral link with 4.5 turn remained an unsolved problem to this date, even though the graphs that admit the double crossover polyhedral links with 4.5 turn have been synthesized. In this paper, we provide a complete formulation of the braid index of a double crossover polyhedral link with an arbitrary turn number. Our approach is more general and it allows us to completely determine the braid indices for a much larger class of links. In the case of the double crossover polyhedral links, our formulation of the braid index is a simple formula based on a simpler graph used as a template to build the double crossover polyhedral links.
Highlights
The synthesis of topologically interesting structures like braids in the range of nanometer to micrometer is becoming popular
A fundamental topological invariant that is sometimes used to describe the complexity of a molecule, is another potential tool that can be used to study the complexity properties of certain DNA molecules, some of which have been synthesized in laboratories by chemists and biologists in recent years
Through four arm immobile DNA crossover junctions, the following DNA polyhedral links with polyhedral shapes have been synthesized in laboratories: DNA cube [4], DNA tetrahedron [5], DNA octahedron [6], DNA truncated octahedron [7], DNA bipyramid [8], DNA dodecahedron [9], and DNA dodecahedron and buckyballs [10]
Summary
The synthesis of topologically interesting structures like braids in the range of nanometer to micrometer is becoming popular. We present a solution to this problem as the consequence of a much more general result, that is, we present a solution that would allow us to determine the braid index of a double crossover polyhedral link with an arbitrary turn number. The main result of this paper is the determination of the braid index for any link in a large class of positive (or negative) alternating links This link class includes all double crossover polyhedral links with any given turn number. In addition to providing the determination of an important measure for characterizing and analyzing the structure and complexity of DNA polyhedra modeled by the double crossover polyhedral links (with any given turn number), our research can be used as tools in the study of topological entanglement of more general biopolymers encountered in DNA nanotechnology. We end the paper by showing how our approach and results in this paper may be used to provide alternative proofs for some previously known results in the case of double crossover polyhedral links with 4 turn, as well as obtaining some parallel (new) results for the case of double crossover polyhedral links with 4.5 turn
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