  Polymer chains composing a polymer solution in strict two dimensions (2D) are
characterized with irregular domain boundaries, whose fractal dimension
($\mathcal{D}^{\partial}$) varies with the area fraction of the solution and
the solvent quality. {\color{black}Our analysis of numerical simulations of
polymer solutions finds} that $\mathcal{D}^{\partial}$ in good solvents changes
non-monotonically from $\mathcal{D}^{\partial}=4/3$ in dilute phase to
$\mathcal{D}^{\partial}=5/4$ in dense phase, maximizing to
$\mathcal{D}^{\partial}\approx 3/2$ at a crossover area fraction $\phi_{\rm
cr}\approx 0.2$, whereas for polymers in $\Theta$ solvents
$\mathcal{D}^{\partial}$ remains constant at $\mathcal{D}^{\partial}=4/3$ from
dilute to semi-dilute phase. Using polymer physics arguments, we rationalize
these values, and show that the maximum irregularity of
$\mathcal{D}^\partial\approx 3/2$ is due to "fjord"-like corrugations formed
along the domain boundaries which also maximize at the same crossover area
fraction. Our finding of $\mathcal{D}^\partial\approx 3/2$ is, in fact, in
perfect agreement with the upper bound for the fractal dimension of the
external perimeter of 2D random curves at scaling limit, which is predicted by
the Schramm-Loewner evolution (SLE).
