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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 programto interpret the colored Jones polynomial or a knot in the context of SU(2)gauge theory. This paper explains why there are $\mathbb{R}$-invariantsolutions 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 solutionsis 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 thepaper, there is a $\mathbb{C}^{(m_0 -m_\infty - 2)/2}\times \mathbb{C}^*$moduli space of these interpolation solutions.)

The existence of instanton solutions to the $\mathbb{R}$-invariant Kapustin-Witten equations on $(0,\infty)\times \mathbb{R}^2\times \mathbb{R}$