The elastic constant anisotropy and the core structure of wedge disclinations with strengths s = +/-1/2 of lyotropic liquid crystals of a soluble polydiacetylene, P-4-BCMU (($) over bar M(w) = 5.15 X 10(5), ($) over bar M(n) = 2.16 X 10(5), and ($) over bar M(w)/($) over bar M(n) = 2.4),in chloroform was studied. The Frank elastic constant anisotropy defined by epsilon = (k(11) - k(33))/(k(11) + k(33)) for this polymer was determined by three different methods. The results show that the value of epsilon for this polymer is 0.5 (+/-0.05) in solid state and/or in liquid crystal state, indicating that the splay constant k(11) is three time higher than the bend constant k(33). This result further implies that splay is unfavorable in the liquid crystal as it requires greater energy than bend. In the area adjacent to the core of a singularity the value of epsilon increases with decreasing distance to the core, indicating increasing anisotropy of the elastic constants toward the core. The influence of the elastic constant anisotropy on the optical texture of the LC polymer when viewed by polarizing light microscope is studied.