Graded knot polynomials

Most knot polynomial invariants are not polynomial time computable. But after deforming the knot polynomial and dividing it according to certain grading, each component can be polynomial time computable. In this way, knot invariants that were previously very complex can now be calculated with their low-order components. This idea was applied to HOMFLY polynomial by mathematicians such as Jozef H. Przytycki, F. Jaeger, Akio Kawauchi and others. This report generalizes this result, showing that the knot polynomial deformation they give is a special case of the reporter’s knot invariant, and discusses more general possibilities.