The architecture of Sierpinski knots

Match

Volumn 68, Issue 2, 2012, Pages 595-610

  Duan, Jin Wei ^a   Zheng, Zhen ^a   Zhou, Pan Pan ^a   Qiu, Wen Yuan ^a  

^a LANZHOU UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84885833610     PISSN: 03406253     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (59)

1. W. Y. Qiu, Z. Wang, G. Hu, The Chemistry and Mathematics of DNA Polyhedra, Nova, New York, 2010.
2. A. J. Turberfield, DNA as an engineering material, Phys. World 16 (2003) 43-46.
3. N. C. Seeman, Nanotechnology and the double helix, Sci. Amer. 290 (2004) 64-75.
4. R. P. Goodman, R. M. Berry, A. J. Turberfield, The single-step synthesis of a DNA tetrahedron, Chem. Commun. 12 (2004) 1372-1373.
5. R. P. Goodman, A. T. Schaap, C. F. Tardin, C. M. Erben, R. M. Berry, C. F. Schmidt, A. J. Turberfield, Rapid chiral assembly of rigid DNA building blocks for molecular nanofabrication, Science 310 (2005) 1661-1665.
6. Y. He, T. Ye, M. Su, C. Zhang, A. E. Ribbe, W. Jiang, C. D. Mao, Hierarchical self-assembly of DNA into symmetric supramolecular polyhedral, Nature 452 (2008) 198-202.
7. C. Zhang, M. Su, Y. He, Y. J. Leng, A. E. Ribbe, G. S. Wang, W. Jiang, C. D. Mao, Exterior modification of a DNA tetrahedron, Chem. Commun. 46 (2010) 6792-6794.
8. R. P. Goodman, M. Heilemann, S. Doose, C. M. Erben, A. N. Kapanidis, A. J. Turberfield, Reconfigurable, braced, three dimensional DNA nanostructures, Nat. Nanotechnol. 3 (2008) 93-96.
9. Y. G. Ke, J. Sharma, M. H. Liu, K. Jahn, Y. Liu, H. Yan, Scaffolded DNA origami of a DNA tetrahedron molecular container, Nano Lett. 9 (2009) 2445-2447.
10. Z. Li, B. Wei, J. Nangreave, C. X. Lin, Y. Liu, Y. L. Mi, H. Yan, A replicable tetrahedral nanostructure self-assembled from a single DNA strand, J. Am. Chem. Soc. 131 (2009) 13093-13098.
11. J. Chen, N. C. Seeman, Synthesis from DNA of a molecule with the connectivity of a cube, Nature 350 (1991) 631-633.
12. N. C. Seeman, DNA in a material world, Nature 421 (2003) 427-431.
13. C. Zhang, S. H. Ko, M. Su, Y. J. Leng, A. E. Ribbe, W. Jiang, C. D. Mao, Symmetry controls the face geometry of DNA polyhedra, J. Am. Chem. Soc. 131 (2009) 1413-1415.
14. Y. Zhang, N. C. Seeman, Construction of a DNA-truncated octahedron, J. Am. Chem. Soc. 116 (1994) 1661-1669.
15. W. M. Shih, J. D. Quispe, G. F. Joyce, A 1.7-kilobase single-stranded DNA that folds into a nanoscale octahedron, Nature 427 (2004) 618-621.
16. F. F. Andersen, B. Knudsen, C. L. P. Oliveira, R. F. Frøhlich, D. Krüger, J. Bungert, M. Agbandje-McKenna, R. McKenna, S. Juul, C. Veigaard, J. Koch, J. L. Rubinstein, B. Guldbrandtsen, M. S. Hede, G. Karlsson, A. H. Andersen, J. S. Pedersen, B. R. Knudsen, Assembly and structural analysis of a covalently closed nano-scale DNA cage, Nucleic Acids Res. 36 (2008) 1113-1119.
17. Y. He, M. Su, P. A. Fang, C. Zhang, A. E. Ribbe, W. Jiang, C. D. Mao, On the chirality of self-assembled DNA octahedra, Angew. Chem. Int. Ed. 48 (2009) 748-751.
18. J. Zimmermann, M. P. J. Cebulla, S. Münninghoff, G. von Kiedrowski, Self-assembly of a DNA dodecahedron from 20 trisoligonucleotides with C3h linkers, Angew. Chem. Int. Ed. 47 (2008) 3626-3630.
19. C. Zhang, M. Su, Y. He, X. Zhao, P. A. Fang, A. E. Ribbe, W. Jiang, C. D. Mao, Conformational flexibility facilitates self-assembly of complex DNA nanostructures, Proc. Natl. Acad. Sci. U. S. A. 105 (2008) 10665-10669.
20. D. Bhatia, S. Mehlab, R. Krishnan, S. S. Indi, A. Basu, Y. Krishnan, Icosahedral DNA nanocapsules by modular assembly, Angew. Chem. Int. Ed. 48 (2009) 4134-4137.
21. C. M. Erben, R. P. Goodman, A. J. Turberfield, A self-assembled DNA bipyramid, J. Am. Chem. Soc. 129 (2007) 6992-6993.
22. N. C. Seeman, DNA components for molecular architecture, Acc. Chem. Res. 30 (1997) 357-363.
23. C. D. Mao, W. Q. Sun, N. C. Seeman, Assembly of Borromean rings from DNA, Nature 386 (1997) 137-138.
24. F. A. Aldaye, H. F. Sleiman, Modular access to structurally switchable 3D discrete DNA assemblies, J. Am. Chem. Soc. 129 (2007) 13376-13377.
25. A. J. Mastroianni, S. A. Claridge, A. P. Alivisatos, Pyramidal and chiral groupings of gold nanocrystals assembled using DNA scaffolds, J. Am. Chem. Soc. 131 (2009) 8455-8459.
26. H. Dietz, S. M. Douglas, W. M. Shih, Folding DNA into twisted and curved nanoscale shapes, Science 325 (2009) 725-730.
27. E. S. Andersen, M. D. Dong, M. M. Nielsen, K. Jahn, R. Subramani, W. Mamdouh, M. M. Golas, B. Sander, H. Stark, C. L. P. Oliveira, J. S. Pedersen, V. Birkedal, F. Besenbacher, K. V. Gothelf, J. Kjems, Self-assembly of a nanoscale DNA box with a controllable lid, Nature 459 (2009) 73-76.
28. A. Kuzuya, M. Komiyama, Design and construction of a box-shaped 3D-DNA origami, Chem. Commun. 28 (2009) 4182-4184.
29. W. Y. Qiu, Z. Wang, G. Hu, The chemistry and mathematics of DNA polyhedra, in: W. I. Hong (Ed.), Mathematical Chemistry, Chemistry Research and Applications Series, Nova, New York, 2010, pp. 327-366.
30. P. W. K. Rothemund, Folding DNA to create nanoscale shapes and patterns, Nature 440 (2006) 297-302.
31. D. Rolfsen, Knots and Links, Publish or Perish, Berkely, 1976.
32. C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, Freeman, New York, 1994.
33. W. Y. Qiu, Knot theory, DNA topology, and molecular symmetry breaking, in: D. Bonchev, D. H. Rouvray (Eds.), Chemical Topology. Applications and Techniques, Gordon and Breach, Amsterdam, 2000, pp. 175-237.
34. W. Y. Qiu, X. D. Zhai, Molecular design of Goldberg polyhedral links, J. Mol. Struc. (Theochem) 756 (2005) 163-166.
35. W. Y. Qiu, X. D. Zhai, Y. Y. Qiu, Architecture of Platonic and Archimedean polyhedral links, Sci. China Ser. B: Chemistry 1 (2008) 13-18.
36. D. Lu, G. Hu, Y. Y. Qiu, W. Y. Qiu, Topological transformation of dual polyhedral links, MATCH Commun. Math. Comput. Chem. 63 (2010) 67-78.
37. G. Hu, W. Y. Qiu, Extended Goldberg polyhedral links with odd tangles, MATCH Commun. Math. Comput. Chem. 61 (2009) 753-766.
38. G. Hu, W. Y. Qiu, Extended Goldberg polyhedral links with even tangles, MATCH Commun. Math. Comput. Chem. 61 (2009) 737-752.
39. G. Hu, W. Y. Qiu, X. S. Cheng, S. Y Liu, The complexity of Platonic and Archimedean polyhedral links, J. Math. Chem. 48 (2010) 401-412.
40. G. Hu, X. D. Zhai, D. Lu, W. Y. Qiu, The architecture of Platonic polyhedral links, J. Math. Chem. 46 (2009) 592-603.
41. X. A. Jin, F. J. Zhang, Zeros of the Jones polynomial for multiple crossing-twisted links, J. Stat. Phys. 140 (2010) 1054-1064.
42. X. A. Jin, F. J. Zhang, The Homfly polynomial for even polyhedral links, MATCH Commun. Math. Comput. Chem. 63 (2010) 657-677.
43. X. A. Jin, F. J. Zhang, Jones polynomials and their zeros for a family of links, Phys. A 333 (2004) 183-196.
44. http://math.ict.edu.rs:8080/webMathematica/poly/cont.htm
45. S. Jablan, L. Radović, R. Sazdanović, Pyramidal knots and links and their invariants, MATCH Commun. Math. Comput. Chem. 65 (2011) 541-580.
46. S. Jablan, L. Radović, R. Sazdanović, Knots and links derived from prismatic graphs, MATCH Commun. Math. Comput. Chem. 66 (2011) 65-92.
47. X. S. Cheng, W. Y. Qiu, H. P. Zhang, A novel molecular design of polyhedral links and their chiral analysis, MATCH Commun. Math. Comput. Chem. 62 (2009) 115-130.
48. X. S. Cheng, S. Y. Liu, H. P. Zhang, W. Y. Qiu, Fabrication of a family of pyramidal links and their genus, MATCH Commun. Math. Comput. Chem. 63 (2010) 623-636.
49. X. S. Cheng, H. P. Zhang, G. Hu, W. Y. Qiu, The Architecture and Jones Polynomials of cycle-crossover polyhedral links, MATCH Commun. Math. Comput. Chem. 63 (2010) 637-656.
50. S. Y. Liu, X. S. Cheng, H. P. Zhang, W. Y. Qiu, The architecture of polyhedral links and their HOMFLY polynomials, J. Math. Chem. 48 (2010) 439-456.
51. S. Y. Liu, H. P. Zhang, W. Y. Qiu, The HOMFLY Polynomial for a family of polyhedral links, MATCH Commun. Math. Comput. Chem. 67 (2012) 65-90.
52. P. W. K. Rothemund, N. Papadakis, E. Winfree, Algorithmic self-assembly of DNA Sierpinski triangles, PLoS Biol. 2 (2004) 2041-2053.
53. J. W. Duan, Z. Zheng, P. P. Zhou, W. Y. Qiu, The architecture of DNA Sierpinski links, MATCH Commun. Math. Comput. Chem. 67 (2012) 817-832.
54. http://en.wikipedia.org/wiki/Sierpinski-triangle
55. H. Seifert, Über das geschlecht von knoten, Math. Annalen. 110 (1934) 571-592.
56. D. Gabai, Genera of the alternating links, Duke Math. J. 53 (1986) 677-681.
57. D. S. Richeson, Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton Univ. Press, Princeton, 2008.
58. P. R. Cronwell, Knots and Links, Cambridge Univ. Press, Cambridge, 2004.
59. N. C. Seeman, Nucleic acid junctions and lattices, J. Theor. Biol. 99 (1982) 237-247.