Knot polynomial invariants in classical Abelian Chern-Simons field theory

TL;DR

This paper demonstrates that classical abelian Chern-Simons field theory naturally produces knot polynomial invariants such as the Kauffman polynomial and Jones polynomial, linking topological quantum invariants with classical field theory.

Contribution

It introduces a topological invariant derived from abelian Chern-Simons action that reproduces known knot polynomials via skein relations, bridging classical field theory and knot invariants.

Findings

Derived a topological invariant from abelian Chern-Simons theory.

Reproduced Kauffman R-polynomial and bracket polynomial skein relations.

Computed bracket polynomials for trefoil knots and constructed the Jones polynomial.

Abstract

Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant $t^{I\left( \mathcal{L} \right) }$ is constructed for a link $\mathcal{L}$, where $I$ is the abelian Chern-Simons action and $t$ a formal constant. For oriented knotted vortex lines, $t^{I}$ satisfies the skein relations of the Kauffman R-polynomial; for un-oriented knotted lines, $t^{I}$ satisfies the skein relations of the Kauffman bracket polynomial. As an example the bracket polynomials of trefoil knots are computed, and the Jones polynomial is constructed from the bracket polynomial.