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

  • Meilian Li,

    Affiliations School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, P. R. China, School of Mathematics and Computer Sciences, Longyan University, Longyan, Fujian, P. R. China

  • Qingying Deng,

    Affiliation School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, P. R. China

  • Xian’an Jin

    xajin@xmu.edu.cn

    Affiliation School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, P. R. China

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

  • Meilian Li, 
  • Qingying Deng, 
  • Xian’an Jin
PLOS
x

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.

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 [710]. 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.

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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.

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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

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 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.

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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

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; FY = FY(1 − ω) denotes the flow polynomial in variable ω of ⟨Y⟩, the spanning subgraph of G with edge set Y; πU denotes the product of the labels of the edges in U = EY.

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 aω.

(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, ⋯, as 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, ⋯, as by a single edge a. Then Ch(H) can be obtained from Ch(G) by replacing a1 a2as by a and conversely, Ch(G) can be obtained from Ch(H) by replacing a by a1 a2as.

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

Lemma 2 ([39], The Y − △ theorem) Let GY 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 GY by a Y − △ transformation, where the rest of G is the same as in GY, as shown in Fig 7. Then

Lemma 2 implies that Ch(G) can be obtained from Ch(GY) = 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 46 (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, ⋯, vn and m edges labeled a1, a2, ⋯, am. 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 Uij is a subset of cardinality i of {a1, a2, ⋯, am} and j=1,2,,(mi); Yij denotes the complementary subset of Uij. Then we can obtain Ch(G′), namely (2) where pij and qij are the numbers of k’s such that qk = 3 and qk = 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.

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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 vi and vj are the end-vertices of the edge labeled ak. Recall that where Uij is a subset of cardinality i of {a1, a2, ⋯, am} and j=1,2,,(mi); Yij denotes the complementary subset of Uij. By Lemma 1, replace ak by bi abj in Ch(G), we obtain Ch(G*). For a fixed i from 0 to m, UijFYijπUij becomes aiUijFYijk=1nbkqk, where qk is the number of edges in Uij incident with the vertex vk. Thus we have: (3) where qk = 0, 1, 2, 3 depending on Uij. Additionally, for nonzero terms in Ch(G*), qk can not be 2. When ql = 0, it means that bl doesn’t appear in k=1nbkqk.

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 bk3 (namely qk = 3) by bk3ω and bk by bk2+bkω1, multiplying the other terms (namely, the term V in Lemma 2) by 3bkω − 2. We divide it into two steps for clarity.

Step 2. For each k = 1, 2, ⋯, n, for each term of Ch(G*), replacing bk3 by x and bk by y, we obtain a polynomial in x, y, a, ω, denoted it by Ch(G**), namely, (4) where pij and qij are the numbers of k’s such that qk = 3 and qk = 1 in Ch(G*), respectively. Note that pm1 = n, qm1 = 0, p01 = 0 and q01 = 0.

Step 3. In Ch(G**), replace x by b3ω3bω2 and y by b2+bω13bω2, and normalize entire polynomial by (3bω − 2)n.

Therefore, the first term ami=1nbi3 becomes am(b3ω)n and the last term FG becomes FG(3bω − 2)n. Note that the numbers of k’s such that qk = 0 is npijqij, which is exactly the times we need multiply the term corresponding to Uij by 3bω − 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 Sm 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. a1b1 ab2, a2b1 ab2, a3b1 ab2.

Step 2. b13x,b23x,b1y,b2y

Step 3. xb3w, yb2 + bw − 1, then only need to multiply the last term by (3bω − 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. a1b1 ab3, a2b1 ab4, a3b1 ab2, a4b2 ab4, a5b2 ab3, a6b3 ab4.

Step 2. bk3x,bky,k=1,2,3,4.

Step 3. xb3w, yb2 + bw − 1, then multiplying every term by (3bω − 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.

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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=v1vz. 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 μx(e), μy(e) and wx(e), wy(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 v3v12z2(v+v1)z3(1+v2)2 and −z−1(v + v3)−1, respectively. Combining it with the relation between the dichromatic polynomial QP 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 = K1K2 ∪ ⋯ ∪ Kn be an oriented link with n components. Let A(L) = (aij)n×n be the linking matrix with aij = lk(Ki, Kj), the linking number of Ki and Kj, if ij and aii=j=1,jinlk(Ki,Kj). 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 PL(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 spanv PL(v, z) as follows: (5) where spanv PL(v, z) = max degv PL(v, z) − min degv PL(v, z), and max degv PL(v, z) and min degv PL(v, z) denote, respectively, the maximal degree and minimal degree of v in the polynomial PL(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. 1. Chen J, Seeman NC. Synthesis from DNA of a molecule with the connectivity of a cube. Nature. 1991;350: 631–633. pmid:2017259
  2. 2. Goodman RP, Berry RM, Turberfield AJ. The single-step synthesis of a DNA tetrahedron. Chem. Commun. 2004;12: 1372–1373. pmid:15179470
  3. 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
  4. 4. Zhang Y, Seeman NC. The construction of a DNA truncated octahedron. J. Am. Chem. Soc. 1994;116: 1661–1669.
  5. 5. Erben CM, Goodman RP, Turberfield AJ. A Self-Assembled DNA Bipyramid. J. Am. Chem. Soc. 2007;129: 6992–6993. pmid:17500526
  6. 6. Zimmermann J, Cebulla MPJ, Mönningho S, Kiedrowski GV. Self-Assembly of a DNA Dodecahedron from 20 Trisoligonucleotides, with C3h Linkers. Angew. Chem. Int. Ed. 2008;47: 3626–3630.
  7. 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.
  8. 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
  9. 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
  10. 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
  11. 11. Fu TJ, Seeman NC. DNA Double-Crossover Molecules. Biochemistry. 1993;32: 3211–3220. pmid:8461289
  12. 12. Lin C, Liu Y, Yan H. Designer DNA Nanoarchitectures. Biochemistry. 2009;48: 1663–1674. pmid:19199428
  13. 13. Kuzuya A, Komiyama M. DNA origami: Fold, stick, and beyond. Nanoscale. 2010;2: 310–322. pmid:20644813
  14. 14. Jin XA, Zhang FJ. The Homfly polynomial for even polyhedral links. MATCH Commun. Math. Comput. Chem. 2010;63: 657–677.
  15. 15. Jin XA, Zhang FJ. The Jones polynomial for polyhedral links. MATCH Commun. Math. Comput. Chem. 2011;65: 501–520.
  16. 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.
  17. 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.
  18. 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
  19. 19. Hu G, Qiu WY, Ceulemans A. A New Euler’s Formula for DNA Polyhedra. PLoS ONE. 2011;6(10): e26308. pmid:22022596
  20. 20. Cheng XS, Jiang X, Dai H. The braid index of polyhedral links. J. Math. Chem. 2012;50: 1386–1397.
  21. 21. Cheng XS, Jin XA. The braid index of complicated DNA polyhedral links. PLoS ONE. 2012;7: e48968. pmid:23185287
  22. 22. Cheng XS, Lei YJ, Yang WL. The Homfly polynomial of double crossover links. J. Math. Chem. 2014;52: 23–41.
  23. 23. Duan JW, Hu G, Qiu WY. Topological Aspect of DNA Cages: Genus. MATCH Commun. Math. Comput. Chem. 2014;72(2): 475–488.
  24. 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.
  25. 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.
  26. 26. Zhang H, Zhao S, Lin R. The Forcing Polynomial of Catacondensed Hexagonal Systems. MATCH Commun. Math. Comput. Chem. 2015;73(2): 473–490.
  27. 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.
  28. 28. Przytycki JH, Traczyk P. Invariants of links of Conway type. Kobe J. Math. 1987;4: 115–139.
  29. 29. Jones VFR. A polynomial invariant for knots via Von Neumann algebras. Bull. Amer. Math. Soc. 1985;12: 103–112.
  30. 30. Alexander JW. Topological invariants of knots and links. Tran. Amer. Math. Soc. 1928;30: 275–306.
  31. 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. 32. Cromwell PG. Knots and Links. Cambridge University Press; 2004.
  33. 33. Murasugi K. Knot Theory and Its Applications. Birkhauser; 1996.
  34. 34. Adams CC. The Knot Book. American Mathematical Society; 2004.
  35. 35. Jin XA, Zhang FJ. The Homfly and dichromatic polynomials. Proc. Amer. Math. Soc. 2012;140: 1459–1472.
  36. 36. Jin XA, Zhang FJ. On the location of zeros of the Homfly polynomial. J. Stat. Mech. 2011; P07011.
  37. 37. Read RC, Whitehead EG Jr. Chromatic polynomials of homeomorphism classes of graphs. Discrete Math. 1999;204: 337–356.
  38. 38. Shahmohamad H. A survey on flow polynomial. Utilitas Mathematica. 2002;62: 13–32.
  39. 39. Read RC. Chain polynomials of graphs. Discrete Math. 2003;265: 213–235.
  40. 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.
  41. 41. Hosokawa F. On ∇-polvnomials of links. Osaka Math. J. 1958;10: 273–282.
  42. 42. Hoste J. The first coefficient of the Conway polynomial. Proc. Amer. Math. Soc. 1985;95: 299–302.
  43. 43. Franks J, Williams RF. Braids and the Jones polynomial. Trans. Amer. Math. Soc. 1987;303: 97–108.
  44. 44. Morton HR. Seifert circles and knot polynomials. Math. Proc. Cambridge Philos. Soc. 1986;99: 107–109.
  45. 45. Ohyama Y. On the minimal crossing number and the braid index of links. Canad. J. Math. 1993;45: 117–131.
  46. 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.
  47. 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