This thesis has six chapters. In Chapters 1 and 2, we give the definitions and examples of the main links under study in this thesis: positive braids links, checkerboard graph links, and basket links. In Chapter 2, we also define the signature function of a link and compute it for the cases of three and four positive Hopf bands plumbed together. In particular, we investigate the behavior of the signature function when the intersection of the core curves of these Hopf bands goes to infinity. In Chapter 3, we define special types of moves for checkerboard graphs and use them to prove that a checkerboard graph link with maximal signature is isotopic to one of the links realized by the simply laced Dinkin diagrams (ADE diagrams). In Chapter 4, we also characterize checkerboard graphs whose corresponding link is of the ADE type and use this to show that there is a linear time algorithm to find these links from a checkerboard graph. In Chapter 5, we study the connection between checkerboard graph links and basket links. In Chapter 6, we prove that a basket link made with positive Hopf bands and symmetrized Seifert form congruent to the Cartan matrix of the simply laced Dynkin diagram A is isotopic to a two-strand torus link. We also provide some examples of fake A links
