This paper studies the topological properties of knotted solitons in the （3 ＋1）-dimensional Aratyn-Ferreira- Zimerman （AFZ） model. Topologically, these solitons are characterized by the Hopf invariant I, which is an integral class in the homotopy group π3（S3） = Z. By making use of the decomposition of U（1） gauge potential theory and Duan＇s topological current theory, it is shown that the invariazat is just the total sum of all the self-linking and linking numbers of the knot family while only linking numbers are considered in other papers. Furthermore, it is pointed out that this invariant is preserved in the branch processes （splitting, merging and intersection） of these knot vortex lines.
An unconventional integer quantum Hall regime was found in magnetic semiconductor-superconductor hybrids. By making use of the decomposition of the gauge potential on a U（1） principal fibre bundle over k-space, we study the topological structure of the integral Hall conductance. It is labeled by the Hopf index β and the Brouwer degree η. The Hall conductance topological current and its evolution is discussed.
Based on Duan＇s topological current theory,we propose a novel approach to study the topological properties of topological defects in a two-dimensional complex vector order parameter system.This method shows explicitly the fine topological structure of defects.The branch processes of defects in the vector order parameter system have also been investigated with this method.
By making use of Duan-Ge＇s decomposition theory of gauge potential and the topological current theory proposed by Prof. Duan Yi-Shi, we study a two-component superfluid Bose condensed system, which is supposed to be realized in the interior of neutron stars in the form of the coexistence of a neutron superfluid and a protonic superconductor. We propose that this system possesses vortex lines. The topological charges of the vortex lines are characterized by the Hopf indices and the Brower degrees of C-mapping.
