  It has been conjectured that the exponential growth rate of the number of
lattice polygons with knot-type $K$ is the same as that for unknot polygons,
and that the entropic critical exponent increases by one for each prime knot in
the knot decomposition of $K$. We provide the first proof of this conjecture by
considering knots and non-split links in tube $\mathbb{T}^*$, the $\infty
\times 2\times 1$ sublattice of the simple cubic lattice. We establish upper
and lower bounds relating the asymptotics of the number of $n$-edge polygons
with fixed link-type in $\mathbb{T}^*$ to that of the number of $n$-edge
unknots. For the upper bounds, we prove that polygons can be unknotted by braid
insertions. For the lower bounds, we prove a pattern theorem for unknots using
information from exact transfer-matrices. We prove new results for 4-plats and
highlight connections to modelling DNA in nanochannels.
