Algebraic and topological structures on rational tangles
DOI:
https://doi.org/10.4995/agt.2017.2250Keywords:
group Hopf algebra, locally finite partial order, tangle, pseudo-module, bi-pseudo-module, pseudo-tensor product, incidence algebra, interval coalgebra, continued fraction, tangle convergent.Abstract
In this paper we present the construction of a group Hopf algebra on the class of rational tangles. A locally finite partial order on this class is introduced and a topology is generated. An interval coalgebra structure associated with the locally finite partial order is specified. Irrational and real tangles are introduced and their relation with rational tangles are studied. The existence of the maximal real tangle is described in detail.
Downloads
References
P. Cartier, A primer of Hopf algebras, Preprint IHES (2006).
J. H. Conway, An enumeration of knots and links and some of their properties, Computational problems in abstract algebra (1986), 329-358.
J. H. Conway and R. K. Guy, Continued fractions, in: The Book of numbers, Springer Verlag (1996).
A. Cuyt, A. B. Petersen, B. Verdonk, H. Waadeland and W. B. Jones, Handbook of continued fractions for special functions, Springer (2008).
I. K. Darcy, Modeling protein-DNA complexes with tangles, Computers and Mathematics with Applications 55 (2008), 924-937.
https://doi.org/10.1016/j.camwa.2006.12.099
J. R. Goldman and L. H. Kauffman, Rational tangles, Adv. Appl. Math. 18 (1997), 300-332.
https://doi.org/10.1006/aama.1996.0511
L. H. Kauffman and S. Lambropoulou, Classifying and applying rational knots and rational tangles, Contemporary Math. 304 (2002), 223-259.
https://doi.org/10.1090/conm/304/05197
L. H. Kauffman and S. Lambropoulou, On the classification of rational knots, Enseign. Math. 49 (2003), 357-410.
L. H. Kauffman and S. Lambropoulou, On the classification of rational tangles, Adv. Appl. Math. 33 (2004), 199-237.
https://doi.org/10.1016/j.aam.2003.06.002
L. Lorentzen and H. Waadeland, Continued fractions, in: Convergence theory, vol. 1, World Scientific (2008).
https://doi.org/10.2991/978-94-91216-37-4
Yu. I. Manin, Quantum groups and noncommutative geometry”, Centre de Recherches Mathematiques (CRM), Universite de Montreal (1991).
J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Annals of Math. 81 (1965), 211-264.
https://doi.org/10.2307/1970615
E. Spiegel and C. J. O'Donnell, Incidence algebras, Marcel Dekker, New York (1997).
M. L. Wachs, Poset topology: tools and applications, Lect. Notes IAS/Parkcity Math. Inst. (2004).
Downloads
Published
How to Cite
Issue
Section
License
This journal is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike- 4.0 International License.
