...what can I do with the darn thing?

Background: I read that still no Vassiliev Invariant with mutant-discriminating
power is known (correct me if this is outdated). Now, my research lead to a
whole load of mutant-discriminating invariants but I preferably work with
*unoriented* knots. (Since my S matrices for unoriented and oriented knots look
rather related, I could also provide with the oriented analogue - I just never
checked whether that is also mutant-discriminating.) Anyway. Here is the S matrix,
it's a rather simple 4D one:

S={{-c^(-1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 0, c*(-c^(-2) + c^2), 0, 0, 0, 0, 0, 0, 0, 0, -c, 0, 0, 0},

{0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 0, c^(-3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 0, 0, c^3, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0},

{0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 0, c^3, 0, 0, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, c^(-3), 0, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0},

{0, 0, 0, -c, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0},

{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -c^(-1)}};

cap=cup={{0, 0, 0, (-I)*c}, {0, 0, 1, 0}, {0, 1, 0, 0}, {I/c, 0, 0, 0}};
writhe normalization=c^3, c is a free parameter.

Now question 1: For example, if I have the skein relation for the Alexander polynomial, I can do a lot of tangle calculus with that until I arrive at a skein relation where I can reverse an arrow and the equation still holds, thus effectively landing in my familiar unoriented knot area again. Could this also work for some Vassiliev invariant of sufficient high order (or a linear combination of several VI)? The snag is of course VI are defined via Gauss diagrams, not skein relations, and I have no idea whether "reversing one arrow" does make sense at all in that context.

Question 2: How do I check *practically* if a certain S matrix (say,
for oriented knots - I have tons of them either) is also a VI?
OK, it has to vanish for n doublepoints but how do I actually
*compute* that (I can only prove that S is *not* a VI by computing
the value of a random n-doublepoint knot - if the result is nonzero,
fine, but if not, I must try another...and another...Does a result
exist that a degree n VI is proven as such if you just check some f(n)
"independent" knots?)

(Soft) Question 3 of course is my first sentence. Is having a mutant-discriminating
invariant of good *practical* use for knot taxonomists? After all, there
is more than one way to distinguish two knots...