A Proposed Correspondence Between NP-Completeness and the Mandelbrot Set via Knot Theory

The P versus NP question is one of the most profound open problems in theoretical computer science, with implications across mathematics, cryptography, and complexity theory. This work proposes a structured framework establishing a correspondence between the solvability of NP-complete problems, the topological triviality of knots, and the dynamical stability of points in the complex plane. Using the Partition Problem as a case study, we introduce a deterministic “dynamic weaving” algorithm that maps problem instances to braid representations, which are then evaluated using the Jones polynomial. The braids are generated from the orbital paths of points c under the Mandelbrot iteration \( z \mapsto z^2 + c \), with stability conjectured to correspond to solvability. Computational experiments, implemented in SageMath with SnapPy, demonstrate a consistent mapping between solvable instances and the unknot, and unsolvable instances and topologically complex links. The principal open challenge—the derivation of a universal mapping S → c—is formulated precisely, and we outline how its resolution could yield a new class of complexity-theoretic invariants rooted in topology and dynamical systems.