A General Method for Computing the Homfly Polynomial of DNA Double Crossover 3-Regular Links
May
A General Method for Computing the Homfly Polynomial of DNA Double Crossover 3- Regular Links
Meilian Li 0 1 2
Qingying Deng 0 1 2
Xian 0 1 2
an Jin 0 1 2
0 1 School of Mathematical Sciences, Xiamen University , Xiamen, Fujian , P. R. China , 2 School of Mathematics and Computer Sciences, Longyan University , Longyan, Fujian , P. R. China
1 Funding: This work was supported by J 11271307 National Natural Science Foundation of China and J 2012J01019 Natural Science Foundation of Fujian Province of China. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript
2 Academic Editor: Yongtang Shi, Nankai University , CHINA
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.
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Competing Interests: The authors have declared
that no competing interests exist.
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. For example, the DNA
cube [1], DNA tetrahedron [2], DNA octahedron [3], DNA truncated octahedron [4], DNA
bipyramid [5] and DNA dodecahedron [6]. In recent several years, a type of more complicated
DNA polyhedra have been reported in [710]. They are all synthesized by the strategy of
npoint stars. In fact they are called double crossover DNA polyhedra in [11]. In addition,
similar DNA molecular structures can also be found in [12, 13]. Polyhedral links modelling the
double crossover DNA polyhedra are called double crossover polyhedral links. As an example,
the planar diagram of the double crossover hexahedral link is given in Fig 1.
The DNA double crossover hexahedron was assembled from two different component
three-point-star tiles (A and B), the process is shown in Fig 2. The hexahedral structures have
been confirmed by multiple techniques including polyacrylamide gel electrophoresis (PAGE),
dynamic light scattering (DLS), cryogenic electron microscopy (cryo-EM) imaging, and single
particle three-dimensional (3D) reconstruction [9]. We shall use the orientation of the 2
backbone strands of the dsDNA to orient DNA polyhedral links. Thus we always consider DNA
polyhedral links as oriented links with antiparallel orientations. Under this orientation, the
double crossover hexahedral link in Fig 1 is a negative one, i.e., each crossing is left-handed.
See Fig 3.
For understanding, describing and quantizing DNA polyhedra, many invariants of
polyhedral links modeling DNA polyhedra have been computed and analyzed [1426]. Among these
invariants, the Homfly polynomial [27, 28] is a very powerful one. It bears much information
of oriented links, containing the Jones polynomial [29] and Alexander-Conway polynomial
[30, 31] as special cases. The Homfly polynomial can distinguish most links from their mirror
images, and it helps to determine other numerical invariants such as braid index and the genus
etc [3234]. Unfortunately, computing the Homfly polynomial is, in general, very hard.
Computer software (e.g. KnotGTK) can only deal with links with small (about 50) number
of crossings.
Mathematically, given any planar (not necessarily polyhedral) graph, we can construct the
corresponding double crossover link by covering the vertex of degree n with the n-point star.
In this paper we shall focus on 3-regular, i.e. cubic plane graphs and call the corresponding
double crossover links the double crossover 3-regular links. 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 46 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
Fig 2. Assembly of DNA 4-turn hexahedra from two different component three-point-star tiles (A and B).
doi:10.1371/journal.pone.0125184.g002
Fig 4. The labeled theta graph and the labeled triangular prism truncated from the label (...truncated)