Seifert Matrix for a Knot's Mirror Image

Question

Question: What is the relation between the Seifert matrix of a knot, $V$, found using some given Seifert surface, $S$, and the Seifert matrix for its mirror image, $V'$, found using the mirror image of the given Seifert surface, $S'$?

The mirror image can be formed by switching every crossing of the Seifert matrix (i.e. every under-crossing in $S$ becomes an over-crossing in $S'$ and vice versa). When forming the mirror image, the orientation is reversed as well.

My thoughts so far: I am not sure what happens to a general entry $V_{i,j}$ when we form the mirror image. My initial thought is that $V_{i,j}=-V'_{j,i}$. My reason for believing this comes from the Seifert matrix for the trefoil and its mirror image.

I have calculated the Seifert matrix for the trefoil: $$\begin{bmatrix}-1 & 1\\0 & -1\end{bmatrix}$$ as well as the Seifert matrix for the mirror image of the trefoil: $$\begin{bmatrix}1 & 0\\-1 & 1\end{bmatrix}$$

Any help would be greatly appreciated.

Answer 1

I take it you are reversing the orientation of the Seifert surface when taking the mirror image. The orientation of the Seifert surface induces the orientation of the knot, so you are computing the Seifert matrix of the mirror image of the knot with reversed orientation. Separately, the Seifert matrix of the mirror image of the knot is $-V$ and the Seifert matrix for the knot with reversed orientation is $V^T$. (Proposition 6.12 of Lickorish.)

Briefly:

• When doing a mirror image, all the crossings flip, and $\operatorname{lk}(f_i^{-},f_j)$ becomes negative what it was, where $f_i^{-}$ is the push-off of $f_i$ behind the Seifert surface.
• When reversing orientation, the push-off is backwards from what it was, and is $\operatorname{lk}(f_i^{+},f_j)$ with respect to the original orientation, which is equivalently $\operatorname{lk}(f_i,f_j^{-})$. Thus the Seifert matrix is transposed.