Let $P:=P(a\_1,\dots,a\_n)$ be a Pretzel link ( <https://en.wikipedia.org/wiki/Pretzel_link> ).
For every permutation $\sigma\in S\_n$ we can consider the link $$\sigma P:=P(a\_{\sigma(1)},\dots,a\_{\sigma(n)})$$. If $\sigma=(12\dots n)^k$ for some k then $P$ and $\sigma P$ are equivalent.
My questions are:


* Are there any results related to the general case?
* Which invariants can be used to distinguish pretzel links which differ only by a permutation of their coefficients?


Note that $\sigma P$ is a mutant of $P$ for every $\sigma\in S\_n$. So this problem is related to a more general problem i.e. how to distinguish mutant knots (or links).



How to distinguish Pretzel links with the same coefficients?
[Richard Bedient](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-112/issue-2/Double-branched-covers-and-pretzel-knots/pjm/1102709604.full) has proved in 1984 that two Pretzel knots $P$ and $\sigma P$ are equivalent if and only if $\sigma$ is a cyclic permutation, an order reversing permutation, or a composition of both. He computes a particular group constructed from the fundamental group of the complement to distinguish every non-equivalent pair of knots.


Finding invariants is non-obvious since mutant knots have the same Jones (and HOMFLY) polynomial and hyperbolic volume. These particular mutant knots also have the same branched double covering (a Seifert manifold fibering over $S^2$ with singular fibers of order $a\_1, \ldots, a\_n$). In fact, Bedient was interested in finding families of non-equivalent knots sharing the same branched double covering.


**Edit.**  I missed some quite restricting hypothesis on the coefficients. The theorem holds if the $a\_i$'s are distinct and all strictly greater than $1$. Moreover we must have $\sum\_i 1/a\_i <1$.


As noted by Ian, such a theorem cannot hold in general. For instance, a consecutive pair $(+1,-1)$ can be moved everywhere.


