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.
KeywordsDNA polyhedron Polyhedral links Homfly polynomial Chain polynomial
X.-S. Cheng was supported by Grants from the National Natural Science Foundation of China (No. 11101174), Natural Science Foundation of Guangdong Province of China (No. S2011040003984) and Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. LYM11120). Y. Lei and W. Yang was supported by Grants from the National Natural Science Foundation of China (No. 11271307), Natural Science Foundation of Fujian Province of China (No. 2012J01019).
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