  For general Temperley-Lieb loop models, including the logarithmic minimal
models ${\cal LM}(p,p')$ with $p,p'$ coprime integers, we construct an infinite
family of Robin boundary conditions on the strip as linear combinations of
Neumann and Dirichlet boundary conditions. These boundary conditions are
Yang-Baxter integrable and allow loop segments to terminate on the boundary.
Algebraically, the Robin boundary conditions are described by the one-boundary
Temperley-Lieb algebra. Solvable critical dense polymers is the first member
${\cal LM}(1,2)$ of the family of logarithmic minimal models and has loop
fugacity $\beta=0$ and central charge $c=-2$. Specializing to ${\cal LM}(1,2)$
with our Robin boundary conditions, we solve the model exactly on strips of
arbitrary finite size $N$ and extract the finite-size conformal corrections
using an Euler-Maclaurin formula. The key to the solution is an inversion
identity satisfied by the commuting double row transfer matrices. This
inversion identity is established directly in the Temperley-Lieb algebra. We
classify the eigenvalues of the double row transfer matrices using the physical
combinatorics of the patterns of zeros in the complex spectral parameter plane
and obtain finitized characters related to spaces of coinvariants of
$\mathbb{Z}_4$ fermions. In the continuum scaling limit, the Robin boundary
conditions are associated with irreducible Virasoro Verma modules with
conformal weights $\Delta_{r,s-\frac{1}{2}}=\frac{1}{32}(L^2-4)$ where
$L=2s-1-4r$, $r\in\mathbb{Z}$, $s\in\mathbb{N}$. These conformal weights
populate a Kac table with half-integer Kac labels. Fusion of the corresponding
modules with the generators of the Kac fusion algebra is examined and general
fusion rules are proposed.
