Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 141 / 1 / e

Star the unbounded region $r_0$ and the adjacent lobe region $a$. The remaining columns of the row-scaled matrix are $(b,c,d)$: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 entriesTheir signed determinant product is $-1$, one summand inThe other two states give $t$ and $-t^2$, and the total differs from $t^2-t+1$ only by the unit $-1$.