Structure of symmetry group of some composite links and some applications
Keywords:knot, link, geometric topology, symmetry group, classification of links
In this paper, we study the symmetry group of a type of composite topological links, such as 22m#22 . We have done a complete analysis on the elements of the symmetric group of this link and show the structure of the group. The results can be generalized to the study of the symmetry group of any composite topological link, and therefore it can be used for the classification of composite topological links, which can also be potentially used to identify synthetics molecules.
S. Akbulut and H. King, All knots are algebraic, Commentarii Mathematici Helvetici 56, no. 1 (1981), 339-351. https://doi.org/10.1007/BF02566217
J. C. Álvarez Paiva and A. C. Thompson, Volumes on normed and Finsler spaces, Riemann-Finsler Geometry, MSRI Publications 49 (2004), 1-46.
M. F. Atiyah, The geometry and physics of knots, Cambridge University Press, 1990. https://doi.org/10.1017/CBO9780511623868
A. Bernig, Valuations with crofton formula and finsler geometry, Advances in Mathematics 210, no. 2 (2007), 733-753. https://doi.org/10.1016/j.aim.2006.07.009
J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, in: Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), pages 329-358, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5
P. R. Cromwel, Knots and links, Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511809767
I. K. Darcy, Biological distances on dna knots and links: applications to xer recombination, Journal of Knot Theory and its Ramifications 10, no. 2 (2001), 269-294. https://doi.org/10.1142/S0218216501000846
D. S. Dummit and R. M. Foote, Abstract algebra, volume 1999, Prentice Hall Englewood Cliffs, NJ, 1991.
M. H. Freedman, R. E. Gompf, S. Morrison and K. Walker, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, Quantum Topology 1, no. 2 (2010), 171-208. https://doi.org/10.4171/QT/5
M.-L. Ge and Ch. N. Yang, Braid group, knot theory and statistical mechanics, World Scientific, 1989.
D. Gorenstein, R. Lyons and R. Solomon, The classification of finite simple groups, volume 1, Plenum Press New York, 1983. https://doi.org/10.1007/978-1-4613-3685-3_1
L. H. Kauffman, Knots and physics, volume 53, World scientific, 2013. https://doi.org/10.1142/8338
X.-S. Lin, Z. Wang, et al., Integral geometry of plane curves and knot invariants, J. Differential Geom. 44, no. 1 (1996), 74-95. https://doi.org/10.4310/jdg/1214458740
Y. Liu, Ropelength under linking operation and enzyme action, General Mathematics 16, no. 1 (2008), 55-58.
Y. Liu, On the range of cosine transform of distributions for torus-invariant complex Minkowski spaces, Far East Journal of Mathematical Sciences 39, no. 2 (2010), 733-753.
Y. Liu, On the explicit formula of Holmes-Thompson areas in integral geometry, preprint.
M. W. Scheeler, D. Kleckner, D. Proment, G. L Kindlmann and W. T. M. Irvine, Helicity conservation by flow across scales in reconnecting vortex links and knots, Proceedings of the National Academy of Sciences 111, no. 43 (2014), 15350-15355. https://doi.org/10.1073/pnas.1407232111
A. Stasiak, V. Katritch and L. H. Kauffman, Ideal Knots, Series on Knots and Everything, Vol. 19, World Scientific, Singapore, 1998. https://doi.org/10.1142/3843
D. W. Sumners, Untangling Dna, The Mathematical Intelligencer 12, no. 3 (1990), 71-80. https://doi.org/10.1007/BF03024022
D. W. Sumners, The knot theory of molecules, Journal of mathematical chemistry 1, no. 1 (1987), 1-14. https://doi.org/10.1007/BF01205335
S. A. Wasserman, J. M. Dungan and N. R. Cozzarelli, Discovery of a predicted DNA knot substantiates a model for site-specific recombination, Science 229, no. 4709 (1985), 171-174. https://doi.org/10.1126/science.2990045
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