Dual graphs and knot invariants Original Research Article

Let G be a signed plane graph and Gd its signed dual graph. Methods from knot theory are used to show that the signed Laplacian matrices L(G) and L(Gd) are Goeritz congruent. There exists diagonal (0,±1)-matrices, Δ1 and Δ2, and a unimodular matrix U such that image Δ1circled plusL(G)=U(Δ2circled plusL(Gd))Ut.For Laplacian matrices of an unsigned plane graph and its dual, L(G) is Goeritz congruent to −L(Gd).