In dealing with quantum groups, q-deformed integers play a prominent role, capturing the notion of graded dimension. Morier-Genoud and Ovsienko recently introduced the concept of q-deformed rationals: a rational number r/s gets deformed into a rational fraction in the variable q. This definition, originally related to the notion of continued fraction expansion, has attracted quite a lot of attention, and connections have been made in particular to the Burau representation of braids. I will start by giving definitions closer to that latter, topological world. Then I will present ongoing work with Perrine Jouteur, where we use these q-rationals in the context of web categories, providing interpolating families of categories that should be thought of as deformations of Deligne's Rep(Gl(t)) categories.