Knots and links derived from prismatic graphs

Match

Volumn 66, Issue 1, 2011, Pages 65-92

  Jablan, Slavik ^a   Radović, Ljiljana ^b   Sazdanović, Radmila ^a  

^a MATHEMATICAL INSTITUTE   (Serbia)

^b UNIVERSITY OF NI   (Serbia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 80155214125     PISSN: 03406253     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Review

References (47)

1. S. Jablan, LJ. Radovic, R. Sazdanovic, Pyramidal knots and links and their invariants, MATCH Commun. Math. Comput. Chem. 65 (2011) 541-580.
2. W. Y. Qiu, X. D. Zhai, Y. Y. Qiu, Architecture of Platonic and Archimedean polyhedral links, Set. China, Ser. B: Chemistry 51 (2008) 13-18.
3. Y. M. Yang, W. Y. Qiu, Molecular design and mathematical analysis of carbon nanotube links, MATCH Commun. Math. Comput. Chem. 58 (2007) 635-646.
4. S. V. Jablan, R. Sazdanović, LinKnot - Knot Theory by Computer, World Scientific, New Jersey, 2007.
5. W. Y. Qiu, X. D. Zhai, Molecular design of Goldberg polyhedral links, J. Mol. Struc. (Theochem) 756 (2005) 163-166.
6. G. Hu, W. Y. Qiu, Extended Goldberg polyhedra, MATCH Commun. Math. Comput. Chem. 59 (2008) 585-594.
7. G. Hu, W. Y. Qiu, Extended Goldberg polyhedral links with even tangles, MATCH Commun. Math. Comput. Chem. 61 (2009) 737-752.
8. G. Hu, W. Y. Qiu, Extended Goldberg polyhedral links with odd tangles, MATCH Commun. Math. Comput. Chem. 61 (2009) 753-766.
9. 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.
10. 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.
11. X. S. Cheng, H. Zhang, G. Hu, W. Y. Qiu, The architecture and Jones polynomials of cycle-crossover polyhedral links, MATCH Commun. Math. Comput. Chem. 63 (2010) 637-654.
12. X. Jin, F. Zhang, The Kauffman brackets for equivalence classes of links, Adv. Appl. Math. 34 (2005) 47-64.
13. X. Jin, F. Zhang, The replacements of signed graphs and Kauffman brackets of link families, Adv. Appl. Math. 39 (2007) 155-172. (Pubitemid 46783167)
14. S. Y. Liu, X. S. Cheng, H. Zhang, W. Y. Qiu, The architecture of polyhedral links and their Homfly polynomials, J. Math. Chem. 48 (2010) 439-456.
15. X. Jin, F. Zhang, The Homfly polynomial for even polyhedral links, MATCH Commun. Math. Comput. Chem. 63 (2010) 657-677.
16. 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.
17. D. Lu, W. Y. Qiu, The keeping and reversal of chirality for dual links, MATCH Commun. Math. Comput. Chem. 63 (2010) 79-90.
18. F. Jaeger, Tutte polynomials and link polynomials, Proc. Amer. Math. Soc. 103 (1988) 647-654.
19. J. H. Chen, N. C. Seeman, Synthesis from DNA of a molecule with the connectivity of a cube, Nature 350 (1991) 631-633. (Pubitemid 21912289)
20. N. C. Seeman, DNA in a material world, Nature 421 (2003) 427-431. (Pubitemid 36157949)
21. Y. W. Zhang, N. C. Seeman, Construction of a DNA-truncated octahedron, J. Am. Chem. Soc. 116 (1994) 1661-1669.
22. 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. (Pubitemid 38248477)
23. C. M. Erben, R. P. Goodman, A. J. Turberfield, A self-assembled DNA bipyramid, J. Am. Chem. Soc. 129 (2007) 6992-6993. (Pubitemid 46903253)
24. F. A. Aldaye, H. F. Sleiman, Modular acess to structurally switchable 3D discrete DNA assemblies, J. Am. Chem. Soc. 129 (2007) 13376-13377. (Pubitemid 350071751)
25. J. Conway, An enumeration of knots and links and some of their related properties, in: J. Leech (Ed.) Computational Problems in Abstract Algebra, Pergamon Press, New York, 1970, pp. 329-358.
26. A. Caudron, Classification des nceuds et des enlancements, Public. Math. d'Orsay 82. Univ. Paris Sud, Dept. Math., Orsay, 1982.
27. W. B. R. Lickorish, An Introduction to Knot Theory, Springer, New York, 1991.
28. K. Murasugi, Knot Theory and its Applications, Birkhäuser, Boston, 1996.
29. S. V. Jablan, R. Sazdanović, Braid family representatives, J. Knot Theory Ramif. 17 (2008) 817-833.
30. M. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987) 297-309.
31. L. H. Kauffman, A Tutte polynomial for signed graphs, Discr. Appl. Math. 25 (1989) 105-127.
32. B. Bollobás, Modern Graph Theory, Springer-Verlag, Berlin, 1998.
33. G. Haggard, D. J. Pearce, G. Royle, Computing Tutte polynomials, preprint.
34. S. V. Jablan, L. Radović, R. Sazdanović, Tutte and Jones polynomial of link families, Filomat 24 (2010) 19-33.
35. S. C. Chang, R. Shrock, Zeros of Jones polynomials for families of knots and links, Physica A 301 (2001) 196-218. (Pubitemid 33089161)
36. F. Y. Wu, J. Wang, Zeroes of Jones polynomial, Physica A 296 (2001) 483-494.
37. X. S. Lin, Zeroes of Jones polynomial, preprint.
38. X. Jin, F. Zhang, Zeroes of the Jones polynomials for families of pretzel links, Physica A 328 (2003) 391-408.
39. X. Jin, F. Zhang, F. Dong, E. G. Tay, On zeroes of the Jones polynomial, Combin. Graph Theory, preprint.
40. Z. X. Tao, On the roots of Jones polynomial, Proceedings of the Second East Asian School of Knots and Related Topics in Geometric Topology, Dalian, China, 2005.
41. The On-Line Encyclopedia of Integer Sequences, http://www.research.att. com/~njas/sequences/
42. F. Harary, E. M. Palmer, Graphical Enumeration, Academic Press, New York, 1973, pp. 33-50.
43. 3, Am. Math. Soc, Providence, 2002, pp. 223-259.
44. H. F. Trotter, Non-invertible knots exist, Topology 2 (1963) 275-280.
45. J. Weeks, A. C. Manoharan, SnapPea PC, http://sites.google.com/a/mota. stanford.edu/snappea-pc-for-windows/
46. M. Culler, N. Dunfield, SnapPy, http://www.math.uic.edu/t3m/SnapPy/doc/
47. L. H. Kauffman, S. V. Jablan, A theorem on amphicheiral alternating links, preprint.