Probabilistic pseudo knot theory

We develop the theory of \emph{probabilistic pseudo knots}, providing a framework for modeling knot diagrams with unresolved crossing information. Pseudo knot diagrams generalize classical diagrams by allowing certain crossings to remain unspecified; in the probabilistic setting, each such \emph{pre-crossing}, namely a crossing with undetermined over--under information, is assigned a probability describing the likelihood of resolving as a positive crossing, with complementary probability assigned to the negative resolution. This induces a probability distribution on complete classical resolutions and, by aggregation, a distribution on classical knot types, capturing uncertainty arising in physical, biological, and computational contexts. We introduce \emph{probabilistic equivalence}, defined via total variation distance between resolution distributions, and extend classical numerical quantities such as writhe and linking number to this setting. We also develop new probabilistic constructions, including the probabilistic chirality index, minimal resolution genus, probabilistic Seifert surface distributions, and polynomial invariants extending the Kauffman bracket. We further discuss matrix-based constructions, including probabilistic Seifert and Goeritz-type matrices, as well as probabilistic surgery producing distributions over 3-manifolds. Finally, we discuss potential applications in molecular biology, materials science, and computational topology.