Knots play a fundamental role in the dynamics of biological and physical systems, from DNA to turbulent plasmas, as well as in climbing, weaving, sailing, and surgery. Despite having been studied for centuries, the subtle interplay between topology and mechanics in elastic knots remains poorly understood. Here, we combined optomechanical experiments with theory and simulations to analyze knotted fibers that change their color under mechanical deformations. Exploiting an analogy with long-range ferromagnetic spin systems, we identified simple topological counting rules to predict the relative mechanical stability of knots and tangles, in agreement with simulations and experiments for commonly used climbing and sailing bends. Our results highlight the importance of twist and writhe in unknotting processes, providing guidance for the control of systems with complex entanglements.