Dual graphs and knot invariants

Abstract

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 thatΔ1⊕L(G)=U(Δ2⊕L(Gd))Ut.For Laplacian matrices of an unsigned plane graph and its dual, L(G) is Goeritz congruent to −L(Gd).