A link is called **quasipositive** if it has a special braid diagram, namely a product of conjugates of the positive standard generators of the braid group. If this product only contains words of the form $\sigma\_{i,j} = (\sigma\_i \cdots \sigma\_{j-2})\sigma\_{j-1} ( \sigma\_i \cdots \sigma\_{j-2})^{-1}$ then we call the link **strongly quasipositive**. These notions were introduced by Rudolph. 



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> Is every quasipositive knot strongly quasipositive? 
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On pp.102 of Rudolph's [Quasipositivity and new knot invariants](http://www.mat.ucm.es/serv/revmat/vol2-1/vol2-1d.pdf), he gives an example of a quasipositive **link** (of 3 components) which is not strongly quasipositive. Is there an example of a **knot** which is quasipositive but not strongly quasipositive?



Is every quasipositive knot strongly quasipositive?
As pointed out by [Hedden](http://arxiv.org/abs/math/0509499), [Livingston](http://arxiv.org/abs/math/0311036) showed that strongly quasipositive knots have $g(K) = g\_4(K) = \tau(K)$, where $g\_4$ is the smooth slice genus and $\tau$ is the Ozsváth-Szabó concordance invariant, so this is an easy way to show that some quasipositive knots aren't strongly quasipositive. [Baader](http://arxiv.org/abs/math/0504594) gave a list of knots which are quasipositive but not positive up to 10 crossings, and among these the criterion $g = g\_4$ is actually equivalent to strong quasipositivity, so for example $8\_{20}$ (with $g=1$ but $g\_4=0$) is quasipositive but not strongly quasipositive.


