The Jones polynomial from a Goeritz matrix

We give an explicit algorithm for calculating the Kauffman bracket of a link diagram from Goeritz matrix that link. Further, we show how Jones polynomial can be recovered when corresponding checkerboard surface is orientable, or more information known about its Gordon–Litherland form. In process develop theory matrices cographic matroids, which extends to any symmetric integer matrix. place this work in context links thickened surfaces.