Algebraic and topological structures on rational tangles

[EN] 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.Milani, V.; Mansourbeigi, SM.; Finizadeh, H. (2017). Algebraic and topological structures on rational tangles. Applied General Topology. 18(1):1-11. doi:10.4995/agt.2017.2250.111181P. Cartier, A primer of Hopf algebras, Preprint IHES (2006).Darcy, I. K. (2008). Modeling protein–DNA complexes with tangles. Computers & Mathematics with Applications, 55(5), 924-937. doi:10.1016/j.camwa.2006.12.099Goldman, J. R., & Kauffman, L. H. (1997). Rational Tangles. Advances in Applied Mathematics, 18(3), 300-332. doi:10.1006/aama.1996.0511Kauffman, L. H., & Lambropoulou, S. (2002). Classifying and applying rational knots and rational tangles. Contemporary Mathematics, 223-259. doi:10.1090/conm/304/05197Kauffman, L. H., & Lambropoulou, S. (2004). On the classification of rational tangles. Advances in Applied Mathematics, 33(2), 199-237. doi:10.1016/j.aam.2003.06.002Lorentzen, L., & Waadeland, H. (2008). Continued Fractions. Atlantis Studies in Mathematics for Engineering and Science. doi:10.2991/978-94-91216-37-4Milnor, J. W., & Moore, J. C. (1965). On the Structure of Hopf Algebras. The Annals of Mathematics, 81(2), 211. doi:10.2307/197061