capacities give sharp conditions under which an ellipsoid of type $E(a, b)$ embeds in a polydisk $P(\nu,\mu).$

### The Nontrivial bundle

One would expect results similar to those of Theorem \[thm:MainTheoremTrivial\] to hold for the twisted bundle $M_{\mu}^{1}$. However, it turns out there is no essential difference between odd and even widths. Instead, all the complexity appears at the special value $k=8$.

\[thm:MainTheoremTwistedK=8\] There exist three functions $u_{1}(\mu,n)$, $u_{2}(\mu,n)$, $u_{3}(\mu,n)$, depending on $\mu\in(0,\infty)$ and $n\in{\mathbb{Z}}\setminus{0}$, all linear in $\mu$, such that $${\mathsf{w}}_{8}\left(M_{\mu}^{1}\right) = \min_{n\in{\mathbb{Z}}\setminus\{0\}} 
\left\{
c_{\operatorname{vol}}=\sqrt{\frac{2\mu+1}{8}},~u_{1}(\mu,n),~u_{2}(\mu, n),~u_{3}(\mu,n),~\frac{6\mu+6}{17}
\right\}$$

In Section \[section:TwistedBundle\], we give explicit formulae for the functions $u_{i}(\mu,n)$, as well as complete descriptions of the generalized Gromov width ${\mathsf{w}}_{8}$ as a piecewise linear functions of $\mu$. As a corollary, we show that there exist infinitely values of $\mu$ for which we can fully pack the nontrivial bundle with $8$ disjoint balls. Indeed, if we define the set $\mathcal{S}\subset(0,\infty)$ by setting $$\mathcal{S} = \left\{\frac{8n^{2}-8n+1}{16n^{2}},~\frac{1}{2},~\frac{8n^{2} + 8n + 1}{16n^{2}}\right\},~n\geq 1$$ then we have

There is a full packing of the nontrivial bundle $M_{\mu}^{1}$ by $8$ balls if, and only if, $\mu\in\mathcal{S}$.

The general case $k\geq 9$ is easier to deal with as the number of obstructions is always finite, namely

\[thm:MainTheoremTwisted\] 