  A non-negative integer labeled set of model solutions to the
$\mathbb{R}$-invariant Kapustin-Witten equations on $(0,\infty)\times
\mathbb{R}^2\times \mathbb{R}$ plays a central role in Edward Witten's program
to interpret the colored Jones polynomial or a knot in the context of SU(2)
gauge theory. This paper explains why there are $\mathbb{R}$-invariant
solutions to these equations on $(0,\infty)\times \mathbb{R}^2\times
\mathbb{R}$ that interpolate between two model solutions as the $(0,\infty)$
parameter increases from 0 to $\infty$ while respecting the $\mathbb{R}^2$
factor asymptotics. The only constraint on the limiting pair of model solutions
is this: Letting $m_0$ and $m_\infty$ denote their non-negative integer labels,
then $m_0 - m_\infty$ must be a positive, even integer. (As explained in the
paper, there is a $\mathbb{C}^{(m_0 -m_\infty - 2)/2}\times \mathbb{C}^*$
moduli space of these interpolation solutions.)
