We report on an experimental and theoretical study of a molecularly thin polymer Langmuir layers on the surface of a Stokesian subfluid. Langmuir layers can have multiple phases (fluid, gas, liquid crystal, isotropic or anisotropic solid); at phase boundaries a line tension force is observed. By comparing theory and experiment we can estimate this line tension. We first consider two co-existing fluid phases; specifically a localized phase embedded in an infinite secondary phase. When the localized phase is stretched (by a transient stagnation flow), it takes the form of a bola consisting of two roughly circular reservoirs connected by a thin tether. This shape relaxes to the minimum energy configuration of a circular domain. The tether is never observed to rupture, even when it is more than a hundred times as long as it is thin. We model these experiments by taking previous descriptions of the full hydrodynamics (primarily those of Stone & McConnell and Lubensky & Goldstein), identifying the dominant effects via dimensional analysis, and reducing the system to a more tractable form. The result is a free boundary problem where motion is driven by the line tension of the domain and damped by the viscosity of the subfluid. The problem has a boundary integral formulation which allows us to numerically simulate the tether relaxation; comparison with the experiments allows us to estimate the line tension in the system. We also report on incorporating dipolar repulsion into the force balance and simulating the formation of "labyrinth" patterns.