Narrow Results

We extend the Kauffman state models of the Jones and Alexander polynomials of classical links to state models of their two-variable extensions in the case of singular links. Moreover, we extend both of them to polynomials with d + 1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibility of long singular knots.

In this paper we present the central ideas and results of a rigorous theory of the Chern–Simons functional integral. In particular, we show that it is possible to define the Wilson loop observables (WLOs) for pure Chern–Simons models with base manifold M = ℝ³ rigorously as infinite dimensional oscillatory integrals by exploiting an "axial gauge fixing" and applying certain regularization techniques like "loop-smearing" and "framing". The (values of the) WLOs can be computed explicitly. If the structure group G of the model is Abelian one obtains well-known linking number expressions for the WLOs. If G is Non-Abelian one obtains expressions which are similar but not identical to the state model representations for the Homfly and Kauffman polynomials given in [19, 21, 31].