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.