Why is that the Alexander polynomial cannot detect the chirality In the knot theory lessons (and in wikipedia) it is stated that "The Alexander polynomial cannot detect the chirality", and it may be the case, but what worries me is that it is sometimes deduced right afted showing that $\Delta(K)=\Delta(K^*)$, where $K^*$ is the mirror image of the knot $K$.
But to my logic, it doesn't follow from this, because it can be another property of the Alexander polynomial, for example some structure of its roots or coefficients, that hypothetically can decide that if $\Delta$ has it then $K$ is chiral.
What should be the "proper" explenation of the statement "The Alexander polynomial cannot detect the chirality"?
 A: I consulted KnotInfo to find pairs of knots with the same Alexander polynomial, one with a particular symmetry type and one chiral. An interpretation of the results is that the Alexander polynomial does not detect the symmetry type of a knot.

*

*The knot $8_9$ is fully amphicheiral, and it has the same Alexander polynomial as the chiral knot $11n_{37}$.


*The knot $12a_{341}$ is negative amphicheiral, and it has the same Alexander polynomial as the chiral knot $12a_{126}$.


*The knot $12a_{427}$ is positive amphicheiral, and it has the same Alexander polynomial as the chiral knot $12a_{189}$.


*The knot $5_2$ is reversible, and it has the same Alexander polynomial as the chiral knot $12n_{124}$.
That's not to say the Alexander polynomial contains no information about a knot's symmetry type.  As @plop references in the comments, Theorem 2.4 of this paper has a condition that can sometimes exclude whether a knot is negative amphicheiral.  Recall that the Conway polynomial $\nabla_K(z)$ is related to the Alexander polynomial by $\nabla_K(t^2)=\Delta_K(t)$ (up to the usual factor of $\pm t^{n}$).  If $\nabla_K(z)$ can't factor as $f(z)f(-z)$ for some integer polynomial $f$, then $K$ is not negative amphicheiral (but given the data above, of a chiral knot with the same Alexander polynomial as a negative amphicheiral knot, the converse is not true).

The Jones polynomial has the property that $V(mL;t) = V(L;t^{-1})$, so amphicheiral knots have the same Jones polynomial. Let's do the same exercise for the Jones polynomial to see whether the Jones polynomial detects amphicheirality in general:

*

*$12a_{510}$ is fully amphicheiral, $12a_{821}$ is negative amphicheiral, and $12a_{1231}$ is chiral, but they all have the same Jones polynomial.


*$10_{33}$ is fully amphicheiral, $12n_{610}$ is reversible, and $12n_{278}$ is chiral, but they all have the same Jones polynomial.
It turns out there are no examples in the database of a positive amphicheiral knot with the same Jones polynomial as a chiral or reversible knot.

Regarding a comment by @knotter, here are the two mentioned knots:
14n14148:

14n14150:

I had hoped signature or omega signature might say one is not amphicheiral, but they are indistinguishable that way.  It does turn out you can distinguish them with their Alexander modules.  The second Alexander polynomial for the first knot is $1$, but for the second it is $2t^2-6t+2$.  I'm not sure what properties the higher Alexander polynomials have with respect to symmetry.  (I used https://kmill.github.io/knotfolio/ to compute the Alexander module.  You can copy and paste the diagrams into this program to enter the knots.)
It seems interesting that these knots are Conway mutations of each other.  This is more obvious when 14n14148 is in this form:

The Conway mutation has the effect of taking the mirror image of the 6 crossings on the right half of the diagram.
