= Solution
Star the unbounded region $r_0$ and the adjacent lobe region $a$. The remaining columns of the row-scaled matrix are $(b,c,d)$:
$$
A_{bcd}=
\begin{pmatrix}
1&0&-1\\
t&1&-1\\
0&t&-1
\end{pmatrix}.
$$
Choose at crossings $1,2,3$ the corners in regions $b,c,d$, respectively. Every unstarred region then contains one chosen corner, so this is a <Kauffman state of a knot diagram>. Circle the entries
$$
(A_{bcd})_{1b}=1,
\qquad (A_{bcd})_{2c}=1,
\qquad (A_{bcd})_{3d}=-1.
$$
Their signed determinant product is $-1$, one summand in
$$
\det A_{bcd}=-1+t-t^2.
$$
The other two states give $t$ and $-t^2$, and the total differs from $t^2-t+1$ only by the unit $-1$.
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