A General Method for Computing the Homfly Polynomial of DNA Double Crossover 3-Regular Links

Meilian Li; Qingying Deng; Xian’an Jin

doi:10.1371/journal.pone.0125184

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.

Citation: Li M, Deng Q, Jin X (2015) A General Method for Computing the Homfly Polynomial of DNA Double Crossover 3-Regular Links. PLoS ONE 10(5): e0125184. https://doi.org/10.1371/journal.pone.0125184

Academic Editor: Yongtang Shi, Nankai University, CHINA

Received: December 3, 2014; Accepted: March 10, 2015; Published: May 1, 2015

Copyright: © 2015 Li et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited

Data Availability: All relevant data are within the paper and its Supporting Information files.

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.

Competing interests: The authors have declared that no competing interests exist.

Introduction

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 [7–10]. They are all synthesized by the strategy of “n-point 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.

Download:

Fig 1. The planar graph of a double crossover hexahedral link with 16 × 12 = 192 crossings.

https://doi.org/10.1371/journal.pone.0125184.g001

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.

Download:

Fig 2. Assembly of DNA 4-turn hexahedra from two different component three-point-star tiles (A and B).

https://doi.org/10.1371/journal.pone.0125184.g002

Download:

Fig 3. Right-handed (+) and left-handed crossings (−).

https://doi.org/10.1371/journal.pone.0125184.g003

For understanding, describing and quantizing DNA polyhedra, many invariants of polyhedral links modeling DNA polyhedra have been computed and analyzed [14–26]. 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 [32–34]. 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 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 give a general method to obtain the chain polynomial of a truncated cubic (i.e. 3-regular) graph with two different labels via the chain polynomial of the original cubic graph based on the Y − △ transform theorem in [39]. As a consequence, for example, we obtain the Homfly polynomial of the double crossover hexahedral link with 192 crossings.

Download:

Fig 4. The labeled theta graph Θ and the labeled triangular prism truncated from the labeled theta graph Θ′.

https://doi.org/10.1371/journal.pone.0125184.g004

Download:

Fig 5. The labeled tetrahedral graph T and its truncation T′.

https://doi.org/10.1371/journal.pone.0125184.g005

Download:

Fig 6. The labeled hexahedral graph H and its truncation.

https://doi.org/10.1371/journal.pone.0125184.g006

Method

We rely on two relationships in order to obtain the Homfly polynomial of double crossover 3-regular links. One relation (RI) is between the chain polynomial of a cubic labelled plane graph and that of its truncation with two different labels. See Theorem 1. The other (RII) is between the chain polynomial of truncated cubic graphs with two different labels and the Homfly polynomial of double crossover 3-regular links. See Theorem 2.

1. RI

The chain polynomial was introduced by Read and Whitehead in [37] for studying the chromatic polynomial of homeomorphic class of graphs. A chain in G is a path in which all vertices, except possibly the end vertices, have degree 2 in the graph G. The length of a chain will be the number of edges in it. A graph with edges labeled elements a, b, c, ⋯ of a commutative ring with unity 1 is called a labeled graph. Let G be a labeled graph. We usually identify the edges with their labels for convenience.

Definition 1 The chain polynomial Ch(G) = Ch(G; ω; a, b, c, ⋯) of a labeled graph G is defined as where the summation is over all subsets Y of the edge set E of the graph G; F_Y = F_Y(1 − ω) denotes the flow polynomial in variable ω of ⟨Y⟩, the spanning subgraph of G with edge set Y; π_U denotes the product of the labels of the edges in U = E − Y.

For a survey on the flow polynomial of graphs, see [38].

Proposition 1 ([37]) Let G be a labeled graph. Then

(1) If G has no edges, then Ch(G) = 1.

(2) If G consists of two graphs A and B having at most one vertex in common, then Ch(G) = Ch(A)Ch(B).

(3) The chain polynomial of a loop with the label a is a − ω.

(4) The term independent of the variables a, b, c, ⋯ is the flow polynomial of G.

(5) If a is an edge of G and is not a loop, let H be the graph obtained from G by deleting the edge a, and let K be the graph obtained by contracting it. Then

(i) Ch(G) = (a − 1)Ch(H) + Ch(K).

(ii) Ch(H) is the coefficient of a in Ch(G).

(iii) Ch(K) is obtained from Ch(G) by putting a = 1.

Since the flow polynomial of a graph with bridges is 0, we have:

Lemma 1 Let a₁, a₂, ⋯, a_s be a chain of length s of a labeled graph G. Let H be the labeled graph obtained from G by replacing the chain a₁, a₂, ⋯, a_s by a single edge a. Then Ch(H) can be obtained from Ch(G) by replacing a₁ a₂ ⋯ a_s by a and conversely, Ch(G) can be obtained from Ch(H) by replacing a by a₁ a₂ ⋯ a_s.

Proposition 2 ([39]) Let C be a cut-set of edges in a graph G. Then any term in Ch(G) that contains the labels of all but one of the edges in C has zero coefficient.

In the case of the graph G_Y shown in Fig 7 (left), {x, y, z} is a cut-set. By Proposition 2, in Ch(G_Y) there are no terms containing labels xy except z, yz except x or xz except y.

Download:

Fig 7. The Y − △ transformation.

https://doi.org/10.1371/journal.pone.0125184.g007

Lemma 2 ([39], The Y − △ theorem) Let G_Y be a labeled graph containing a vertex of degree 3 with incident edges labelled x, y and z. Let its chain polynomial be Pxyz + Ax + By + Cz + V. Let G_△ be the labeled graph obtained from G_Y by a Y − △ transformation, where the rest of G_△ is the same as in G_Y, as shown in Fig 7. Then

Lemma 2 implies that Ch(G_△) can be obtained from Ch(G_Y) = Pxyz + Ax + By + Cz + V by the following substitutions:

Let G be a cubic graph, i.e. a 3-regular graph. By truncating G we mean inserting two vertices to each edge of G firstly, then doing the Y − △ transformation to each vertex of degree 3. Let G′ be the truncated graph of G with original edges of G labeled with a and newly produced edges labeled with b. See Figs 4–6 (right). Now we shall provide a general theorem to obtaining Ch(G′) via Ch(G) by substitutions.

Theorem 1 Let G be a cubic graph with n vertices v₁, v₂, ⋯, v_n and m edges labeled a₁, a₂, ⋯, a_m. Let G′ be the truncated graph of G with original edges of G labeled with a and newly produced edges labeled with b. Suppose (1) where U_ij is a subset of cardinality i of {a₁, a₂, ⋯, a_m} and $j = 1, 2, \dots, (\begin{matrix} m \\ i \end{matrix})$ ; Y_ij denotes the complementary subset of U_ij. Then we can obtain Ch(G′), namely (2) where p_ij and q_ij are the numbers of k’s such that q_k = 3 and q_k = 1 in Eq (3).

Proof. We divide the whole proof into three steps.

Step 1. Compute the chain polynomial of the labeled graph G* obtained from the labeled graph G by the replacements shown in Fig 8.

Download:

Fig 8. The construction of the labeled graph G* from the labeled graph G.

https://doi.org/10.1371/journal.pone.0125184.g008

For each k = 1, 2, ⋯, m, we suppose that v_i and v_j are the end-vertices of the edge labeled a_k. Recall that where U_ij is a subset of cardinality i of {a₁, a₂, ⋯, a_m} and $j = 1, 2, \dots, (\begin{matrix} m \\ i \end{matrix})$ ; Y_ij denotes the complementary subset of U_ij. By Lemma 1, replace a_k by b_i ab_j in Ch(G), we obtain Ch(G*). For a fixed i from 0 to m, $\sum_{U_{i j}} F_{Y_{i j}} π_{U_{i j}}$ becomes $a^{i} \sum_{U_{i j}} F_{Y_{i j}} \prod_{k = 1}^{n} b_{k}^{q_{k}}$ , where q_k is the number of edges in U_ij incident with the vertex v_k. Thus we have: (3) where q_k = 0, 1, 2, 3 depending on U_ij. Additionally, for nonzero terms in Ch(G*), q_k can not be 2. When q_l = 0, it means that b_l doesn’t appear in $\prod_{k = 1}^{n} b_{k}^{q_{k}}$ .

In the following, we apply Lemma 2 to each vertex of degree 3 of G*. Namely, for each k = 1, 2, ⋯, n, for each term of Ch(G*), replacing $b_{k}^{3}$ (namely q_k = 3) by $b_{k}^{3} - ω$ and b_k by $b_{k}^{2} + b_{k} - ω - 1$ , multiplying the other terms (namely, the term V in Lemma 2) by 3b_k − ω − 2. We divide it into two steps for clarity.

Step 2. For each k = 1, 2, ⋯, n, for each term of Ch(G*), replacing $b_{k}^{3}$ by x and b_k by y, we obtain a polynomial in x, y, a, ω, denoted it by Ch(G**), namely, (4) where p_ij and q_ij are the numbers of k’s such that q_k = 3 and q_k = 1 in Ch(G*), respectively. Note that p_m1 = n, q_m1 = 0, p₀₁ = 0 and q₀₁ = 0.

Step 3. In Ch(G**), replace x by $\frac{b^{3} - ω}{3 b - ω - 2}$ and y by $\frac{b^{2} + b - ω - 1}{3 b - ω - 2}$ , and normalize entire polynomial by (3b − ω − 2)ⁿ.

Therefore, the first term $a^{m} \prod_{i = 1}^{n} b_{i}^{3}$ becomes a^m(b³ − ω)ⁿ and the last term F_G becomes F_G(3b − ω − 2)ⁿ. Note that the numbers of k’s such that q_k = 0 is n − p_ij − q_ij, which is exactly the times we need multiply the term corresponding to U_ij by 3b − ω − 2.

A polyhedral graph is planar, it is worth pointing out that our Theorem 1 applies to any cubic graphs which are not necessarily planar. Now we provide several examples.

Example 1 The chain polynomial of the labeled theta graph Θ as shown in Fig 4 (left) and the more general labeled generalized theta S_m graph with m > 3 edges are given in [39] and [40]. Note that the triangular prism as shown in Fig 4 (right) is the truncated graph of the theta graph. Applying Theorem 1, we have

Step 1. a₁ → b₁ ab₂, a₂ → b₁ ab₂, a₃ → b₁ ab₂.

Step 2. $b_{1}^{3} \to x, b_{2}^{3} \to x, b_{1} \to y, b_{2} \to y$

Step 3. x → b³ − w, y → b² + b − w − 1, then only need to multiply the last term by (3b − ω − 2)², we obtain: which matches the result in [39].

Example 2 Let T be the tetrahedral graph labeled as shown in Fig 5 (left), whose chain polynomial was calculated in [37, 39]. Applying Theorem 1, we have

Step 1. a₁ → b₁ ab₃, a₂ → b₁ ab₄, a₃ → b₁ ab₂, a₄ → b₂ ab₄, a₅ → b₂ ab₃, a₆ → b₃ ab₄.

Step 2. $b_{k}^{3} \to x, b_{k} \to y, k = 1, 2, 3, 4 .$

Step 3. x → b³ − w, y → b² + b − w − 1, then multiplying every term by (3b − ω − 2)^{4−d(x)−d(y)}, where d(x) and d(y) are degrees of x and y in the corresponding term in Ch(G**), respectively, we obtain: which matches the reuslt in [22].

Example 3 Let H be the labeled hexahedral graph with V(H) = {v₁, v₂, ⋯, v₈} and E(H) = {a₁, a₂, ⋯, a₁₂} as shown in Fig 6 (left). By performing the Maple program in the Appendix of [22]) which can be used to compute the chain polynomial of labelled graph with small number of edges, we obtain the chain polynomial of the labeled hexahedral graph as follows. According to Theorem 1, a simple program in the Maple platform for calculating Ch(G′) from Ch(G) can be written. See S1 Appendix. By applying the program, we obtain the chain polynomial of the truncated hexahedral graph H′ with two labels as shown in Fig 6 (right). Namely,

2. RII

In [22], a relation between the Homfly polynomial of positive double crossover polyhedral link and the chain polynomial of truncated polyhedral graph with two labels is obtained. For completeness we give an outline of the proof of the relation. Here we consider the negative double crossover polyhedral links and the two tangles T₁ which is used to cover the original edge (labelled a) of a polyhedron and T₂ which is used to cover the newly produced edge (labelled b) after truncation are shown in Fig 9.

Download:

Fig 9. The double crossover tangle T₁ (left) and the vertical integer tangle T₂ (right).

https://doi.org/10.1371/journal.pone.0125184.g009

Let T be a 2-tangle. We denote by Nu(T) and De(T) the numerator and denominator of the 2-tangle T, respectively. Let $δ = \frac{v^{- 1} - v}{z}$ . After calculation, we obtain: Hence, After changing v to −v⁻¹, you will find μ(T₁), μ(T₂) and w(T₁), w(T₂) coincide with μ_x(e), μ_y(e) and w_x(e), w_y(e) in Theorem 3.3 of [22].

Let P be a polyhedral graph. Let P′ be the truncated polyhedral graph of P. Let L(P) be the negative double crossover 4-turn link based on P. Then [35] where the weights of the original edges and the newly produced edges by truncation are $\frac{v^{- 3} - v^{- 1} - 2 z^{2} (v + v^{- 1})}{z^{3} {(1 + v^{2})}^{2}}$ and −z⁻¹(v + v³)⁻¹, respectively. Combining it with the relation between the dichromatic polynomial Q_P′ and the chain polynomial Ch(P′) obtained in [36] (see Lemma 2.5), we obtain:

Theorem 2 Let P be a polyhedral graph having x edges. LetP′ be the truncated polyhedral graph of P with two labels a and b (having 2x vertices and 3x edges). Let L(P) be the negative double crossover 4-turn link based on P. In Ch(P′), we let Then

After changing v to −v⁻¹, you can find Theorem 2 coincides with Theorem 3.4 in [22].

Results

In this section 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. To obtain the Homfly polynomial of the positive double crossover polyhedral links, one only need change v to −v⁻¹. Recall that the Conway and Jones polynomials are both special cases of the Homfly polynomial, i.e. Some computational results are too long to be included in this paper, we only list a few terms here:

Thus,

To our surprise their Conway polynomials are so simple, all having only one term. Let L = K₁ ∪ K₂ ∪ ⋯ ∪ K_n be an oriented link with n components. Let A(L) = (a_ij)_n×n be the linking matrix with a_ij = lk(K_i, K_j), the linking number of K_i and K_j, if i ≠ j and $a_{i i} = - \sum_{j = 1, j \neq i}^{n} l k (K_{i}, K_{j})$ . In [41, 42] the authors proved that the first coefficient of ∇_L(z) (i.e. the coefficient of the term of the lowest degree n − 1) is equal to the cofactor of A(L) up to a sign. Our computational results are the same to the results obtained from cofactor of linking matrices.

Let P_L(v, z) be the Homfly polynomial of the oriented link L. In [43] and [44], Franks, Williams and Morton independently gave a lower bound for the braid index b(L) of an oriented link L in terms of span_v P_L(v, z) as follows: (5) where span_v P_L(v, z) = max deg_v P_L(v, z) − min deg_v P_L(v, z), and max deg_v P_L(v, z) and min deg_v P_L(v, z) denote, respectively, the maximal degree and minimal degree of v in the polynomial P_L(v, z). This inequality (5) is usually called MFW inequality. By combining the following result obtained by Ohyama in 1993 [45] which states that for a non-splittable oriented link L, (6) where c(L) is the crossing number of L.

Hence, b(L(θ)) = 25, b(L(T)) = 49 and b(L(H)) = 97. These results coincide with results in [21].

Discussion

A general method is given in this paper 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. Hence, we convert the computation of the chain polynomial of a large graph with two labels to that of the chain polynomial of a small graph. As an application, by combining with the relation between the Homfly polynomial of a double crossover polyhedral link and the chain polynomial of the truncated polyhedral graph with two different labels, we obtain the Homfly polynomial of the double crossover hexahedral link, which has 192 crossings.

From our computational results, we know that the double crossover DNA hexahedral link is topologically chiral and its braid index is 97. More deeply chemical and biological understanding of our computational results deserves further exploring. To our surprise the Conway polynomials of the double crossover links based on the theta graph, the tetrahedral graph and the hexahedral graph all have only one term. It may coincide with Corollary 4.6 in [35].

We only consider double crossover 3-regular links. Similar approach may be developed to deal with double crossover n-regular links. It may be a more difficult task to compute the Homfly polynomial of polyhedral link modeling protein polyhedra in [46, 47].

Supporting Information

S1 Appendix. file details the computational result for the Homfly polynomial of the positive double crossover 4-turn hexahedral link.

https://doi.org/10.1371/journal.pone.0125184.s001

(PDF)

Acknowledgments

This work was started when the first author visited Xiamen University. The authors thank Associate Professor X.-S Cheng, Assistant Professor Weiling Yang, Ms Yujuan Lei and Ms Nana Zhao for some discussions.

Author Contributions

Conceived and designed the experiments: XJ. Performed the experiments: ML QD. Analyzed the data: ML QD. Contributed reagents/materials/analysis tools: XJ. Wrote the paper: ML QD XJ.

References

1. Chen J, Seeman NC. Synthesis from DNA of a molecule with the connectivity of a cube. Nature. 1991;350: 631–633. pmid:2017259
- View Article
- PubMed/NCBI
- Google Scholar
2. Goodman RP, Berry RM, Turberfield AJ. The single-step synthesis of a DNA tetrahedron. Chem. Commun. 2004;12: 1372–1373. pmid:15179470
- View Article
- PubMed/NCBI
- Google Scholar
3. Shih WM, Quispe JD, Joyce GF. A 1.7-kilobase single-stranded DNA that folds into a nanoscale octahedron. Nature. 2004;427: 618–621. pmid:14961116
- View Article
- PubMed/NCBI
- Google Scholar
4. Zhang Y, Seeman NC. The construction of a DNA truncated octahedron. J. Am. Chem. Soc. 1994;116: 1661–1669.
- View Article
- Google Scholar
5. Erben CM, Goodman RP, Turberfield AJ. A Self-Assembled DNA Bipyramid. J. Am. Chem. Soc. 2007;129: 6992–6993. pmid:17500526
- View Article
- PubMed/NCBI
- Google Scholar
6. Zimmermann J, Cebulla MPJ, Mönningho S, Kiedrowski GV. Self-Assembly of a DNA Dodecahedron from 20 Trisoligonucleotides, with C_3h Linkers. Angew. Chem. Int. Ed. 2008;47: 3626–3630.
- View Article
- Google Scholar
7. He Y, Su M, Fang P, Zhang C, Ribbe AE, Jiang W, et al. On the chirality of self-assembled DNA octahedra, Angew. Chem. Int. Ed. 2010;49: 748–751.
- View Article
- Google Scholar
8. He Y, Ye T, Su M, Zhang C, Ribbe AE, Jiang W, et al. Hierarchical self-assembly of DNA into symmetric supramolecular polyhedra. Nature. 2008;452: 198–201. pmid:18337818
- View Article
- PubMed/NCBI
- Google Scholar
9. Zhang C, Ko SH, Su M, Leng Y, Ribbe AE, Jiang W, et al. Symmetry Controls the Face Geometry of DNA Polyhedra. J. Am. Chem. Soc. 2009;131: 1413–1415. pmid:19173666
- View Article
- PubMed/NCBI
- Google Scholar
10. Zhang C, Su M, He Y, Zhao X, Fang P, Ribbe AE, et al. Conformational flexibility facilitates self-assembly of complex DNA nanostructures. PNAS. 2008;105: 10665–10669. pmid:18667705
- View Article
- PubMed/NCBI
- Google Scholar
11. Fu TJ, Seeman NC. DNA Double-Crossover Molecules. Biochemistry. 1993;32: 3211–3220. pmid:8461289
- View Article
- PubMed/NCBI
- Google Scholar
12. Lin C, Liu Y, Yan H. Designer DNA Nanoarchitectures. Biochemistry. 2009;48: 1663–1674. pmid:19199428
- View Article
- PubMed/NCBI
- Google Scholar
13. Kuzuya A, Komiyama M. DNA origami: Fold, stick, and beyond. Nanoscale. 2010;2: 310–322. pmid:20644813
- View Article
- PubMed/NCBI
- Google Scholar
14. Jin XA, Zhang FJ. The Homfly polynomial for even polyhedral links. MATCH Commun. Math. Comput. Chem. 2010;63: 657–677.
- View Article
- Google Scholar
15. Jin XA, Zhang FJ. The Jones polynomial for polyhedral links. MATCH Commun. Math. Comput. Chem. 2011;65: 501–520.
- View Article
- Google Scholar
16. Cheng XS, Qiu WY, Zhang HP. A novel molecular design of polyhedral links and their chiral analysis. MATCH Commun. Math. Comput. Chem. 2009;62: 115–130.
- View Article
- Google Scholar
17. Cheng XS, Liu SY, Zhang HP, Qiu WY. Fabrication of a family of pyramidal links and their genus. MATCH Commun. Math. Comput. Chem. 2010;63: 115–130.
- View Article
- Google Scholar
18. Comoglio F, Rinaldi M. A Topological Framework for the Computation of the HOMFLY Polynomial and Its Application to Proteins. PLoS ONE. 2011;6(4): e18693. pmid:21533239
- View Article
- PubMed/NCBI
- Google Scholar
19. Hu G, Qiu WY, Ceulemans A. A New Euler’s Formula for DNA Polyhedra. PLoS ONE. 2011;6(10): e26308. pmid:22022596
- View Article
- PubMed/NCBI
- Google Scholar
20. Cheng XS, Jiang X, Dai H. The braid index of polyhedral links. J. Math. Chem. 2012;50: 1386–1397.
- View Article
- Google Scholar
21. Cheng XS, Jin XA. The braid index of complicated DNA polyhedral links. PLoS ONE. 2012;7: e48968. pmid:23185287
- View Article
- PubMed/NCBI
- Google Scholar
22. Cheng XS, Lei YJ, Yang WL. The Homfly polynomial of double crossover links. J. Math. Chem. 2014;52: 23–41.
- View Article
- Google Scholar
23. Duan JW, Hu G, Qiu WY. Topological Aspect of DNA Cages: Genus. MATCH Commun. Math. Comput. Chem. 2014;72(2): 475–488.
- View Article
- Google Scholar
24. Sun XB, Zheng X, Qiu WY. The Architecture of DNA Polyhedral Links with Odd Tangles. MATCH Commun. Math. Comput. Chem. 2014;72(2): 489–500.
- View Article
- Google Scholar
25. Zhang H, Deng K. Spectrum of Matching Forcing Numbers of a Hexagonal System with a Forcing Edge. MATCH Commun. Math. Comput. Chem. 2015;73(2): 457–471.
- View Article
- Google Scholar
26. Zhang H, Zhao S, Lin R. The Forcing Polynomial of Catacondensed Hexagonal Systems. MATCH Commun. Math. Comput. Chem. 2015;73(2): 473–490.
- View Article
- Google Scholar
27. Freyd P, Yetter D, Hoste J, Lickorish WBR, Millett K, Ocneanu A. A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. 1985;12: 239–246.
- View Article
- Google Scholar
28. Przytycki JH, Traczyk P. Invariants of links of Conway type. Kobe J. Math. 1987;4: 115–139.
- View Article
- Google Scholar
29. Jones VFR. A polynomial invariant for knots via Von Neumann algebras. Bull. Amer. Math. Soc. 1985;12: 103–112.
- View Article
- Google Scholar
30. Alexander JW. Topological invariants of knots and links. Tran. Amer. Math. Soc. 1928;30: 275–306.
- View Article
- Google Scholar
31. Conway JH. An enumeration of knots and links, and some of their algebraic properties. Comput. Problems in Abstract Algebra. New York: Pergamon Press; 1970;329–358.
32. Cromwell PG. Knots and Links. Cambridge University Press; 2004.
33. Murasugi K. Knot Theory and Its Applications. Birkhauser; 1996.
34. Adams CC. The Knot Book. American Mathematical Society; 2004.
35. Jin XA, Zhang FJ. The Homfly and dichromatic polynomials. Proc. Amer. Math. Soc. 2012;140: 1459–1472.
- View Article
- Google Scholar
36. Jin XA, Zhang FJ. On the location of zeros of the Homfly polynomial. J. Stat. Mech. 2011; P07011.
- View Article
- Google Scholar
37. Read RC, Whitehead EG Jr. Chromatic polynomials of homeomorphism classes of graphs. Discrete Math. 1999;204: 337–356.
- View Article
- Google Scholar
38. Shahmohamad H. A survey on flow polynomial. Utilitas Mathematica. 2002;62: 13–32.
- View Article
- Google Scholar
39. Read RC. Chain polynomials of graphs. Discrete Math. 2003;265: 213–235.
- View Article
- Google Scholar
40. Jin XA, Zhang FJ, Dong FM, Tay EG. Zeros of the Jones polynomial are dense in the complex plane. Electron. J. Comb. 2010;17(1): R94.
- View Article
- Google Scholar
41. Hosokawa F. On ∇-polvnomials of links. Osaka Math. J. 1958;10: 273–282.
- View Article
- Google Scholar
42. Hoste J. The first coefficient of the Conway polynomial. Proc. Amer. Math. Soc. 1985;95: 299–302.
- View Article
- Google Scholar
43. Franks J, Williams RF. Braids and the Jones polynomial. Trans. Amer. Math. Soc. 1987;303: 97–108.
- View Article
- Google Scholar
44. Morton HR. Seifert circles and knot polynomials. Math. Proc. Cambridge Philos. Soc. 1986;99: 107–109.
- View Article
- Google Scholar
45. Ohyama Y. On the minimal crossing number and the braid index of links. Canad. J. Math. 1993;45: 117–131.
- View Article
- Google Scholar
46. Wiko WR, Lilja L, Duda RL, Tsuruta H, Hendrix RW, Johnson JE. Topologically Linked Protein Rings in the Bacteriophage HK97 Capsid. Science. 2000;289: 2129–2133.
- View Article
- Google Scholar
47. Helgstrand C, Wiko WR, Duda RL, Hendrix RW, Johnson JE, Liljas L. The Refined Structure of a Protein Catenane: The HK97 Bacteriophage Capsid at 3.44 A° Resolution. J. Mol. Biol. 2003;334: 885–899. pmid:14643655
- View Article
- PubMed/NCBI
- Google Scholar

[ref1] 1. Chen J, Seeman NC. Synthesis from DNA of a molecule with the connectivity of a cube. Nature. 1991;350: 631–633. pmid:2017259
View Article
PubMed/NCBI
Google Scholar

[2] View Article

[3] PubMed/NCBI

[4] Google Scholar

[ref2] 2. Goodman RP, Berry RM, Turberfield AJ. The single-step synthesis of a DNA tetrahedron. Chem. Commun. 2004;12: 1372–1373. pmid:15179470
View Article
PubMed/NCBI
Google Scholar

[6] View Article

[7] PubMed/NCBI

[8] Google Scholar

[ref3] 3. Shih WM, Quispe JD, Joyce GF. A 1.7-kilobase single-stranded DNA that folds into a nanoscale octahedron. Nature. 2004;427: 618–621. pmid:14961116
View Article
PubMed/NCBI
Google Scholar

[10] View Article

[11] PubMed/NCBI

[12] Google Scholar

[ref4] 4. Zhang Y, Seeman NC. The construction of a DNA truncated octahedron. J. Am. Chem. Soc. 1994;116: 1661–1669.
View Article
Google Scholar

[14] View Article

[15] Google Scholar

[ref5] 5. Erben CM, Goodman RP, Turberfield AJ. A Self-Assembled DNA Bipyramid. J. Am. Chem. Soc. 2007;129: 6992–6993. pmid:17500526
View Article
PubMed/NCBI
Google Scholar

[17] View Article

[18] PubMed/NCBI

[19] Google Scholar

[ref6] 6. Zimmermann J, Cebulla MPJ, Mönningho S, Kiedrowski GV. Self-Assembly of a DNA Dodecahedron from 20 Trisoligonucleotides, with C_3h Linkers. Angew. Chem. Int. Ed. 2008;47: 3626–3630.
View Article
Google Scholar

[21] View Article

[22] Google Scholar

[ref7] 7. He Y, Su M, Fang P, Zhang C, Ribbe AE, Jiang W, et al. On the chirality of self-assembled DNA octahedra, Angew. Chem. Int. Ed. 2010;49: 748–751.
View Article
Google Scholar

[24] View Article

[25] Google Scholar

[ref8] 8. He Y, Ye T, Su M, Zhang C, Ribbe AE, Jiang W, et al. Hierarchical self-assembly of DNA into symmetric supramolecular polyhedra. Nature. 2008;452: 198–201. pmid:18337818
View Article
PubMed/NCBI
Google Scholar

[27] View Article

[28] PubMed/NCBI

[29] Google Scholar

[ref9] 9. Zhang C, Ko SH, Su M, Leng Y, Ribbe AE, Jiang W, et al. Symmetry Controls the Face Geometry of DNA Polyhedra. J. Am. Chem. Soc. 2009;131: 1413–1415. pmid:19173666
View Article
PubMed/NCBI
Google Scholar

[31] View Article

[32] PubMed/NCBI

[33] Google Scholar

[ref10] 10. Zhang C, Su M, He Y, Zhao X, Fang P, Ribbe AE, et al. Conformational flexibility facilitates self-assembly of complex DNA nanostructures. PNAS. 2008;105: 10665–10669. pmid:18667705
View Article
PubMed/NCBI
Google Scholar

[35] View Article

[36] PubMed/NCBI

[37] Google Scholar

[ref11] 11. Fu TJ, Seeman NC. DNA Double-Crossover Molecules. Biochemistry. 1993;32: 3211–3220. pmid:8461289
View Article
PubMed/NCBI
Google Scholar

[39] View Article

[40] PubMed/NCBI

[41] Google Scholar

[ref12] 12. Lin C, Liu Y, Yan H. Designer DNA Nanoarchitectures. Biochemistry. 2009;48: 1663–1674. pmid:19199428
View Article
PubMed/NCBI
Google Scholar

[43] View Article

[44] PubMed/NCBI

[45] Google Scholar

[ref13] 13. Kuzuya A, Komiyama M. DNA origami: Fold, stick, and beyond. Nanoscale. 2010;2: 310–322. pmid:20644813
View Article
PubMed/NCBI
Google Scholar

[47] View Article

[48] PubMed/NCBI

[49] Google Scholar

[ref14] 14. Jin XA, Zhang FJ. The Homfly polynomial for even polyhedral links. MATCH Commun. Math. Comput. Chem. 2010;63: 657–677.
View Article
Google Scholar

[51] View Article

[52] Google Scholar

[ref15] 15. Jin XA, Zhang FJ. The Jones polynomial for polyhedral links. MATCH Commun. Math. Comput. Chem. 2011;65: 501–520.
View Article
Google Scholar

[54] View Article

[55] Google Scholar

[ref16] 16. Cheng XS, Qiu WY, Zhang HP. A novel molecular design of polyhedral links and their chiral analysis. MATCH Commun. Math. Comput. Chem. 2009;62: 115–130.
View Article
Google Scholar

[57] View Article

[58] Google Scholar

[ref17] 17. Cheng XS, Liu SY, Zhang HP, Qiu WY. Fabrication of a family of pyramidal links and their genus. MATCH Commun. Math. Comput. Chem. 2010;63: 115–130.
View Article
Google Scholar

[60] View Article

[61] Google Scholar

[ref18] 18. Comoglio F, Rinaldi M. A Topological Framework for the Computation of the HOMFLY Polynomial and Its Application to Proteins. PLoS ONE. 2011;6(4): e18693. pmid:21533239
View Article
PubMed/NCBI
Google Scholar

[63] View Article

[64] PubMed/NCBI

[65] Google Scholar

[ref19] 19. Hu G, Qiu WY, Ceulemans A. A New Euler’s Formula for DNA Polyhedra. PLoS ONE. 2011;6(10): e26308. pmid:22022596
View Article
PubMed/NCBI
Google Scholar

[67] View Article

[68] PubMed/NCBI

[69] Google Scholar

[ref20] 20. Cheng XS, Jiang X, Dai H. The braid index of polyhedral links. J. Math. Chem. 2012;50: 1386–1397.
View Article
Google Scholar

[71] View Article

[72] Google Scholar

[ref21] 21. Cheng XS, Jin XA. The braid index of complicated DNA polyhedral links. PLoS ONE. 2012;7: e48968. pmid:23185287
View Article
PubMed/NCBI
Google Scholar

[74] View Article

[75] PubMed/NCBI

[76] Google Scholar

[ref22] 22. Cheng XS, Lei YJ, Yang WL. The Homfly polynomial of double crossover links. J. Math. Chem. 2014;52: 23–41.
View Article
Google Scholar

[78] View Article

[79] Google Scholar

[ref23] 23. Duan JW, Hu G, Qiu WY. Topological Aspect of DNA Cages: Genus. MATCH Commun. Math. Comput. Chem. 2014;72(2): 475–488.
View Article
Google Scholar

[81] View Article

[82] Google Scholar

[ref24] 24. Sun XB, Zheng X, Qiu WY. The Architecture of DNA Polyhedral Links with Odd Tangles. MATCH Commun. Math. Comput. Chem. 2014;72(2): 489–500.
View Article
Google Scholar

[84] View Article

[85] Google Scholar

[ref25] 25. Zhang H, Deng K. Spectrum of Matching Forcing Numbers of a Hexagonal System with a Forcing Edge. MATCH Commun. Math. Comput. Chem. 2015;73(2): 457–471.
View Article
Google Scholar

[87] View Article

[88] Google Scholar

[ref26] 26. Zhang H, Zhao S, Lin R. The Forcing Polynomial of Catacondensed Hexagonal Systems. MATCH Commun. Math. Comput. Chem. 2015;73(2): 473–490.
View Article
Google Scholar

[90] View Article

[91] Google Scholar

[ref27] 27. Freyd P, Yetter D, Hoste J, Lickorish WBR, Millett K, Ocneanu A. A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. 1985;12: 239–246.
View Article
Google Scholar

[93] View Article

[94] Google Scholar

[ref28] 28. Przytycki JH, Traczyk P. Invariants of links of Conway type. Kobe J. Math. 1987;4: 115–139.
View Article
Google Scholar

[96] View Article

[97] Google Scholar

[ref29] 29. Jones VFR. A polynomial invariant for knots via Von Neumann algebras. Bull. Amer. Math. Soc. 1985;12: 103–112.
View Article
Google Scholar

[99] View Article

[100] Google Scholar

[ref30] 30. Alexander JW. Topological invariants of knots and links. Tran. Amer. Math. Soc. 1928;30: 275–306.
View Article
Google Scholar

[102] View Article

[103] Google Scholar

[ref31] 31. Conway JH. An enumeration of knots and links, and some of their algebraic properties. Comput. Problems in Abstract Algebra. New York: Pergamon Press; 1970;329–358.

[ref32] 32. Cromwell PG. Knots and Links. Cambridge University Press; 2004.

[ref33] 33. Murasugi K. Knot Theory and Its Applications. Birkhauser; 1996.

[ref34] 34. Adams CC. The Knot Book. American Mathematical Society; 2004.

[ref35] 35. Jin XA, Zhang FJ. The Homfly and dichromatic polynomials. Proc. Amer. Math. Soc. 2012;140: 1459–1472.
View Article
Google Scholar

[109] View Article

[110] Google Scholar

[ref36] 36. Jin XA, Zhang FJ. On the location of zeros of the Homfly polynomial. J. Stat. Mech. 2011; P07011.
View Article
Google Scholar

[112] View Article

[113] Google Scholar

[ref37] 37. Read RC, Whitehead EG Jr. Chromatic polynomials of homeomorphism classes of graphs. Discrete Math. 1999;204: 337–356.
View Article
Google Scholar

[115] View Article

[116] Google Scholar

[ref38] 38. Shahmohamad H. A survey on flow polynomial. Utilitas Mathematica. 2002;62: 13–32.
View Article
Google Scholar

[118] View Article

[119] Google Scholar

[ref39] 39. Read RC. Chain polynomials of graphs. Discrete Math. 2003;265: 213–235.
View Article
Google Scholar

[121] View Article

[122] Google Scholar

[ref40] 40. Jin XA, Zhang FJ, Dong FM, Tay EG. Zeros of the Jones polynomial are dense in the complex plane. Electron. J. Comb. 2010;17(1): R94.
View Article
Google Scholar

[124] View Article

[125] Google Scholar

[ref41] 41. Hosokawa F. On ∇-polvnomials of links. Osaka Math. J. 1958;10: 273–282.
View Article
Google Scholar

[127] View Article

[128] Google Scholar

[ref42] 42. Hoste J. The first coefficient of the Conway polynomial. Proc. Amer. Math. Soc. 1985;95: 299–302.
View Article
Google Scholar

[130] View Article

[131] Google Scholar

[ref43] 43. Franks J, Williams RF. Braids and the Jones polynomial. Trans. Amer. Math. Soc. 1987;303: 97–108.
View Article
Google Scholar

[133] View Article

[134] Google Scholar

[ref44] 44. Morton HR. Seifert circles and knot polynomials. Math. Proc. Cambridge Philos. Soc. 1986;99: 107–109.
View Article
Google Scholar

[136] View Article

[137] Google Scholar

[ref45] 45. Ohyama Y. On the minimal crossing number and the braid index of links. Canad. J. Math. 1993;45: 117–131.
View Article
Google Scholar

[139] View Article

[140] Google Scholar

[ref46] 46. Wiko WR, Lilja L, Duda RL, Tsuruta H, Hendrix RW, Johnson JE. Topologically Linked Protein Rings in the Bacteriophage HK97 Capsid. Science. 2000;289: 2129–2133.
View Article
Google Scholar

[142] View Article

[143] Google Scholar

[ref47] 47. Helgstrand C, Wiko WR, Duda RL, Hendrix RW, Johnson JE, Liljas L. The Refined Structure of a Protein Catenane: The HK97 Bacteriophage Capsid at 3.44 A° Resolution. J. Mol. Biol. 2003;334: 885–899. pmid:14643655
View Article
PubMed/NCBI
Google Scholar

[145] View Article

[146] PubMed/NCBI

[147] Google Scholar

Figures

Abstract

Introduction

Method

1. RI

2. RII

Results

Discussion

Supporting Information

S1 Appendix. file details the computational result for the Homfly polynomial of the positive double crossover 4-turn hexahedral link.

Acknowledgments

Author Contributions

References