multidegree has been computed positively for arbitrary orientations [@BR].

Using Theorems \[thm:multidegree\] and \[thm:Kpoly\], and the rationality of the singularities (from [@BZ]), we give the first formulae for

-   the multidegrees of type $D$ and $E$ quiver loci,

-   the $K$-polynomials for type $A$ quiver loci in non-equioriented cases, and

-   the $K$-polynomials for type $D$ quiver loci.

Unfortunately, our formulae are not positive in the senses of [@KMS; @Bu; @Mi]. Some positivity of the answers is expected on very general grounds (e.g. Theorem D in [@KM]).

The Bott-Samelson crank {#sec:geom}
=======================

The inductive processes in Theorems \[thm:multidegree\] and \[thm:Kpoly\] have their geometric origin in the Bott-Samelson crank [@BS]. Fix a Borel subgroup $B$ with $P\geq B\geq T$. For each simple root $\alpha$ of $G$, let $P_\alpha$ be the corresponding minimal parabolic. Then if $f: C\to Y$ is a $B$-equivariant map, the space $P_\alpha
\times^B C$ has also a natural $B$-equivariant map to $Y$, which we will call $P_\alpha \times^B f$. We call this functor (on the category of $B$-equivariant maps $f:C\to Y$ to a fixed $G$-space) [[****]{}one turn of the Bott-Samelson crank]{}. By projecting onto the first factor, we see that the resulting space is a $C$-bundle over $P_\alpha/B 