Abstract
In the last 20 years or so, chemists and molecular biologists have synthesized some novel DNA polyhedra. Polyhedral links were introduced to model DNA polyhedra and study topological properties of DNA polyhedra. As a very powerful invariant of oriented links, the Homfly polynomial of some of such polyhedral links with small number of crossings has been obtained. However, it is a challenge to compute Homfly polynomials of polyhedral links with large number of crossings such as double crossover 3-regular links considered here. In this paper, a general method is given for computing the chain polynomial of the truncated cubic graph with two different labels from the chain polynomial of the original labeled cubic graph by substitutions. As a result, we can obtain the Homfly polynomial of the double crossover 3-regular link which has relatively large number of crossings.
Highlights
In the last 20 years or so, many DNA biomolecules with the shape of polyhedron have been synthesized by chemists and molecular biologists in the laboratory
A general method is given for computing the chain polynomial of the truncated cubic graph with two different labels from the chain polynomial of the original labeled cubic graph by substitutions
We use Theorem 2 to compute the Homfly polynomial of negative double crossover 3-regular links based on the theta graph, the tetrahedron and the cube
Summary
In the last 20 years or so, many DNA biomolecules with the shape of polyhedron have been synthesized by chemists and molecular biologists in the laboratory. Based on results in [35] and [36], Cheng, Lei and Yang established a relation in [22] between the Homfly polynomial of the double crossover link and the chain polynomial [37] of the truncated graph with two distinct labels (See Figs 4–6 for examples). Using this relation, they obtained the Homfly polynomial of the double crossover tetrahedral link which has 96 crossings. To compute the Homfly polynomial of the double crossover 3-regular link with more large number of crossings, in the paper we
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