I'm trying to prove that the knot $8_{20}$ is quasi-alternating. I thought of using the fact that this knot is actualy aquivalent to the $Pretzel$ knot $P(3,-3,2)$. There's this famous theorem that appears in a papper by Champanerkar and Kaufman and states that $Pretzel$ links $P(p_1,...,p_n,-q)$ with : $ n \geq 1, p_i \geq 1$ for every $ 1 \leq i \leq n $, and $q > min(p_1,...,p_n)$ are quasi-alternating.
My idea is to prove that $ P(3,-3,2) $ and $ P(3,2,-3) $ (or $ P(2,3,-3) $ ) are of the same isotopy type and then use the theorem.
intuitively it makes sens. I tried the $determinant$ and the two variables $Kauffman$ polynomial and It gave the same values for the two knots.
So the question is what is in general the equivalent class of the $Pretzel$ three strand knot $ P(a,b,c) $ ? (for the specific values of $a,b,$ and $c$ for which it is a knot).
THANK YOU IN ADVANCE.
Pretzel knots $K=P(n_1,n_2,\ldots,n_k)$ are equivalent to cycling the $n_i$ and reversing the order. So, $P(n_k,n_1,n_2,\ldots,n_{k-1})$ and $P(n_k,n_{k-1},\ldots,n_2,n_1)$ are all equivalent to $K$. This is not hard to see from the diagrams too. So when $k=3$, this happens to be every possible permutation.
Thus, the three pretzel knots you mention are all equivalent.