This paper presents a novel approach to multi-agent trajectory prediction for autonomous vehicles using braid theory as a fundamental mathematical framework. The method projects future trajectories of interacting agents into braids that precisely describe how trajectories cross over time, enabling exact representation of coordination modes between agents. By fully leveraging the expressivity of the braid representation, the approach conditions predicted trajectories on specific interaction patterns, moving beyond traditional heuristic-based behavior labeling methods. This results in improved joint prediction accuracy while maintaining computational efficiency compared to existing approaches that rely on extensive computational resources.