Achiral-generator conjecture for alternating knot series

Let $K$ be the alternating generator of a series containing an alternating achiral knot. An iterated mutant is a knot obtained through a sequence of mutation operations, and the obverse is the corresponding orientation-reversed or mirrored object. A HOMFLY or Kauffman polynomial is self-conjugate when it is invariant under the relevant conjugation.

Achiral-generator conjecture. Then at least one of the following holds: $K$ is achiral; $K$ is an iterated mutant of its obverse; or $K$ has self-conjugate HOMFLY and/or Kauffman polynomial.

The first two alternatives are stronger than the preceding proved criterion, while the polynomial alternative need not be. The paper presents the assertion as suggested by the proof and gives examples where Gauss sums distinguish an alternating knot from its obverse more effectively than the HOMFLY or Kauffman polynomial.