Topology and knot theory applications in quantum field theory

Topology and knot theory have profoundly influenced modern quantum field theory by revealing deep connections between gauge fields and topological invariants. Topological quantum field theories (TQFTs) disregard the spacetime metric and compute quantities that depend only on the global topology of the manifold. In Chern–Simons theory, observables associated with knotted loops correspond to knot polynomials such as the Jones polynomial. This paper reviews the mathematical framework that relates knots to quantum field theory and develops a methodology for deriving knot invariants from path integrals. We discuss challenges in quantizing metric independent actions and evaluating Wilson loop operators. Our proposed approach derives closed form expressions for linking numbers and demonstrates how perturbative expansions recover known knot invariants. Numerical examples illustrate how different gauge groups and levels affect knot polynomial values. The outcomes highlight the versatility of topological methods in physics and provide insight into potential applications in quantum computation and condensed matter. Our results show that simple mathematical constructs, when interpreted through gauge theory, yield powerful tools for classifying knots and three manifolds.