An operator on regular isotopy invariants of link diagrams (Q1826161)

Let P be a regular isotopy invariant of link diagrams (including the empty diagram) which takes value in a commutative ring \(\Lambda\). For each element \(\lambda\) of \(\Lambda\), \(P+\lambda\) denotes the regular isotopy invariant defined by \((P+\lambda)({\mathcal D})=\sum_{{\mathcal D}'\subset {\mathcal D}}\lambda^{\#({\mathcal D}-{\mathcal D}')}P(\) \({\mathcal D}')\) where \({\mathcal D}'\) ranges all subdiagrams of \({\mathcal D}\) including the empty diagram, and {\#} denotes the number of link components. The author proves that Kauffman's double bracket polynomial can be obtained from a specialization of Kauffman's Dubrovnik polynomial by the above process and vice versa. Thus the 2-parallel version of the Jones polynomial of a link can be expressed in terms of specializations of the Kauffman polynomials of its sublinks, and vice versa.