圈-双交叉多面体链环的Kauffman括号多项式和束多项式
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摘要
多面体链环常常用来解释和刻画DNA和蛋白质多面体.在这篇文章中,我们建立了圈-双交叉链环的KAUFFMAN括号多项式和束多项式之间的联系,不但可以为求解圈-双交叉链环的多项式不变量提供了一种新的方法,还可以为未来的DNA和蛋白质的合成设计提供新的思路.
Polyhedral links are often used to interpret and characterize DNA and protein polyhedra. In this paper, the relationships between the Kauffman bracket polynomial and sheaf polynomial of cycle-double crossover polyhedral links are established, provide not only a new method for solving polynomial invariants of cycle-double crossover links, but also a new idea for the design of DNA and protein synthesis in the future.
Polyhedral links are often used to interpret and characterize DNA and protein polyhedra. In this paper, the relationships between the Kauffman bracket polynomial and sheaf polynomial of cycle-double crossover polyhedral links are established, provide not only a new method for solving polynomial invariants of cycle-double crossover links, but also a new idea for the design of DNA and protein synthesis in the future.
引文
[1] N.C. Seeman. Nanotechnology and the Double Helix[J].Scl Am, 2004, 290(6):64-69.
[2] J.H. Chen, N.C. Seeman. Synthesis from DNA of a molecule with the connectivity of a cube[J]. Nature, 1991, 350(6319):631-363.
[3] M. Scheffler, A. Dorenbeck, S.Jordan, M. Wustefeld, G.Von Kiedrowski.Self-assembly of trisoligonucleotidyls:the case for nano-acetylene and nano-cyclobutadiene[J].Angew Chem. Int. Ed Engl, 1999, 38(22):3311–3315.
[4]C.L.P.Oliveira,S.Juul,H.L.J?rgensen,etal.StructureofNanoscaleTruncatedOctahedralDNACages:Variationof Single-Stranded Linker Regions and Influence on Assembly Yields[J].ACS Nano, 2010, 4(3):1367–1376.
[5] Y. Zhang, N.C. Seeman.Construction of a DNA-truncated octahedron[J].Am. Chem. Soc, 1994, 116(5):1661–1669.
[6] W.M. Shih, J.D. Quispe, G.F. Joyce.A 1.7-kilobase Single-stranded DNA that Folds into a Nanoscale Octahedron[J].Nature,2004, 427(6975):618-621.
[7] F.A. Aldaye, H.F. Sleiman. Modular Access to Structurally Switchable 3D Discrete DNA Assemblies[J]. Am. Chem. Soc,2007,129(44):13376-13377.
[8] C.M. Erben, R.P. Goodman, A.J. Turberfield.A Self-Assembled DNA Bipyramid[J].Am. Chem. Soc, 2007, 129(22):6992-6993.
[9] Y. Ke, J. Sharma, M. Liu, et al.Scaffolded DNA origami of a DNA tetrahedron molecular container[J].Nano Lett, 2009, 9(6):2445-2447.
[10] E.S. Andersen, M. Dong, M. M. Nielsen,et al.Self-assembly of a nanoscale DNA box with a controllable lid[J].Nature, 2009,459(7243):73-76.
[11] A. Kuzuya, M. Komiyama.Design and construction of a box-shaped 3D-DNA origami[J].Chem. Commun, 2009, 28(28):4182-4184.
[12] W.Y. Qiu, X.D. Zhai.Molecular design of Goldberg polyhedral links[J].Mol. Struct.(THEOCHE), 2005,756:163–166.
[13] Y.M. Yang, W.Y. Qiu.Molecular design and mathematical analysis of Carbon Nan-otube Links[J].Math. Comput. Chem,2007, 58:635–646.
[14] X.S. Cheng, W.Y. Qiu, H.P. Zhang.A novel molecular design of polyhedral links and their chiral analysis[J].Math. Comput.Chem, 2009, 62:115–130.
[15] G. Hu, W.Y. Qiu.Extended Goldberg polyhedral links with even tangles[J].Math. Comput. Chem, 2009, 61:737–752.
[16] G. Hu, W.Y. Qiu.Extended Goldberg polyhedral links with even tangles[J].Math. Comput. Chem, 2009, 61:753–766.
[17] W.Y. Qiu, X.D. Zhai, Y.Y. Qiu.Architecture of platonic and archimedean polyhedral links[J].Sci China SerB-Chem, 2008, 51:13–18.
[18] G. Hu, X.D. Zhai, D. Lu, W.Y. Qiu.The architecture of platonic polyhedral links[J].Math. Chem, 2009, 46:592–603.
[19] X.A Jin, F. Zhang.The Kaufiman brackets for equivalence classes of links[J].Math, 2005, 34:47–64.
[20] X.A Jin, F. Zhang.The replacements of signed graphs and Kaufiman brackets of link families[J].Math, 2007, 39:155–172.
[21] X.S. Cheng, H. Zhang, G. Hu, W. Y. Qiu. The architecture and Jones polynomials of cycle-crossover polyhedrallinks[J].MATCH Commun. Math. Comput. Chem, 2010, 63(3):637-656.
[22] L.H. Kauffman.A Tutte polynomial for signed graphs[J].Discrete Appl. Math, 1989, 25:105–127.
[23] L.H. Kaufiman.State models and the Jones polynomial[J].Topology, 1987, 26:395–407.
[24] L.H. Kaufiman.Statistical mechanics and the Jones polynomial[J].AMS Contemp. Math, 1989, 78:263–297.
[25] L.H. Kaufiman.New invariants in knot theory[J].Amer. Math. Monthly, 1988, 95:195–242.
[26] R.C. Read, E.G. Whitehead Jr.Chromatic polynomials of homeomorphism classes of graphs[J].Discrete Math, 1999, 204:337–356.
[27] L. Traldi.A dichromatic polynomial for weighted graphs and link diagrams[J].Math. Soc, 1989, 106:279–286.
[2] J.H. Chen, N.C. Seeman. Synthesis from DNA of a molecule with the connectivity of a cube[J]. Nature, 1991, 350(6319):631-363.
[3] M. Scheffler, A. Dorenbeck, S.Jordan, M. Wustefeld, G.Von Kiedrowski.Self-assembly of trisoligonucleotidyls:the case for nano-acetylene and nano-cyclobutadiene[J].Angew Chem. Int. Ed Engl, 1999, 38(22):3311–3315.
[4]C.L.P.Oliveira,S.Juul,H.L.J?rgensen,etal.StructureofNanoscaleTruncatedOctahedralDNACages:Variationof Single-Stranded Linker Regions and Influence on Assembly Yields[J].ACS Nano, 2010, 4(3):1367–1376.
[5] Y. Zhang, N.C. Seeman.Construction of a DNA-truncated octahedron[J].Am. Chem. Soc, 1994, 116(5):1661–1669.
[6] W.M. Shih, J.D. Quispe, G.F. Joyce.A 1.7-kilobase Single-stranded DNA that Folds into a Nanoscale Octahedron[J].Nature,2004, 427(6975):618-621.
[7] F.A. Aldaye, H.F. Sleiman. Modular Access to Structurally Switchable 3D Discrete DNA Assemblies[J]. Am. Chem. Soc,2007,129(44):13376-13377.
[8] C.M. Erben, R.P. Goodman, A.J. Turberfield.A Self-Assembled DNA Bipyramid[J].Am. Chem. Soc, 2007, 129(22):6992-6993.
[9] Y. Ke, J. Sharma, M. Liu, et al.Scaffolded DNA origami of a DNA tetrahedron molecular container[J].Nano Lett, 2009, 9(6):2445-2447.
[10] E.S. Andersen, M. Dong, M. M. Nielsen,et al.Self-assembly of a nanoscale DNA box with a controllable lid[J].Nature, 2009,459(7243):73-76.
[11] A. Kuzuya, M. Komiyama.Design and construction of a box-shaped 3D-DNA origami[J].Chem. Commun, 2009, 28(28):4182-4184.
[12] W.Y. Qiu, X.D. Zhai.Molecular design of Goldberg polyhedral links[J].Mol. Struct.(THEOCHE), 2005,756:163–166.
[13] Y.M. Yang, W.Y. Qiu.Molecular design and mathematical analysis of Carbon Nan-otube Links[J].Math. Comput. Chem,2007, 58:635–646.
[14] X.S. Cheng, W.Y. Qiu, H.P. Zhang.A novel molecular design of polyhedral links and their chiral analysis[J].Math. Comput.Chem, 2009, 62:115–130.
[15] G. Hu, W.Y. Qiu.Extended Goldberg polyhedral links with even tangles[J].Math. Comput. Chem, 2009, 61:737–752.
[16] G. Hu, W.Y. Qiu.Extended Goldberg polyhedral links with even tangles[J].Math. Comput. Chem, 2009, 61:753–766.
[17] W.Y. Qiu, X.D. Zhai, Y.Y. Qiu.Architecture of platonic and archimedean polyhedral links[J].Sci China SerB-Chem, 2008, 51:13–18.
[18] G. Hu, X.D. Zhai, D. Lu, W.Y. Qiu.The architecture of platonic polyhedral links[J].Math. Chem, 2009, 46:592–603.
[19] X.A Jin, F. Zhang.The Kaufiman brackets for equivalence classes of links[J].Math, 2005, 34:47–64.
[20] X.A Jin, F. Zhang.The replacements of signed graphs and Kaufiman brackets of link families[J].Math, 2007, 39:155–172.
[21] X.S. Cheng, H. Zhang, G. Hu, W. Y. Qiu. The architecture and Jones polynomials of cycle-crossover polyhedrallinks[J].MATCH Commun. Math. Comput. Chem, 2010, 63(3):637-656.
[22] L.H. Kauffman.A Tutte polynomial for signed graphs[J].Discrete Appl. Math, 1989, 25:105–127.
[23] L.H. Kaufiman.State models and the Jones polynomial[J].Topology, 1987, 26:395–407.
[24] L.H. Kaufiman.Statistical mechanics and the Jones polynomial[J].AMS Contemp. Math, 1989, 78:263–297.
[25] L.H. Kaufiman.New invariants in knot theory[J].Amer. Math. Monthly, 1988, 95:195–242.
[26] R.C. Read, E.G. Whitehead Jr.Chromatic polynomials of homeomorphism classes of graphs[J].Discrete Math, 1999, 204:337–356.
[27] L. Traldi.A dichromatic polynomial for weighted graphs and link diagrams[J].Math. Soc, 1989, 106:279–286.