This project explores the mathematical study of knots and links in topology, focusing on differentiating between the two-component Unlink and the Hopf Link using a computational tool named LINKAGE. LINKAGE employs the linking number, calculated through Barycentric Equations, Matrix Algebra, and basic topological principles, to quantify the degree of linking between two closed curves in three-dimensional space. This approach not only distinguishes between different knot structures but also has applications in understanding complex systems such as magnetic field lines in plasma physics. Additionally, this project includes an example where multiple interlinked loops were analyzed over different time stamps using the LINKAGE algorithm. By observing how these links break and evolve, the algorithm demonstrates its ability to track changes in the topological properties of the system. Subsequently, the algorithm was generalized to compute the linking number for any given n-component Hopf link, where n=2,3,4,5, and so on.
The python script for the linkage algorithm is included in the linkage folder.
The python script for the example of interlinked loops is included in the interlinked folder.
The python script for the n-component Hopf link is included in the n-component folder.
Code released under the MIT license.