Given a link $L$, and a diagram $D$ for $L$, a pre-Goeritz matrix $G = [g_{ij}]$ for $L$ is obtained after choosing a checkerboard coloring for $D$ and ordering the white regions. It can be shown that after deleting a row and column from the resulting matrix (thus obtaining a Goeritz matrix) and taking the absolute value of the determinant of the result, one obtains an invariant of the link (namely $\det L = |\Delta_L(-1)|$).
Rather than showing that this is an invariant of the link by seeing that it computes a known invariant, I would like to check directly by showing that the choices do not effect the result and that the result is invariant under the Reidemeister moves. Upon making all of the choices, I have already checked that the Reidemeister moves do not effect the result.
Why is the resulting determinant independent of the choice of row/column that we delete from the pre-Goeritz matrix?
Changing the ordering of the regions just results in swapping the corresponding rows in the Goeritz matrix and therefore does not effect the determinant.
How do I see that inverting the checkerboard does not change the reusult?