Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 141 / 3 / d / Solution

Star the adjacent regions $o,e$. Choose the Kauffman state of a knot diagram putting the marker at $A$ in region $d$, at $B$ in region $a$, at $C$ in region $b$, and at $D$ in region $c$. Every unstarred region is used exactly once, and every crossing has exactly one marker.

In the full Alexander matrix, circle the entries $(A,d),(B,a),(C,b),(D,c)$, namelyIn the reduced Alexander matrix $B$, these are the diagonal entries in column order $(d,a,b,c)$. The associated permutation is the identity, with sign of a permutation $+1$, so this Kauffman state of a knot diagram contributesto $\det B=-t^2+3t-1$. The other four nonzero permutation terms are $t,t,t,-t^2$. The drawing circles the same four entries literally; the boxes above identify them in the typeset matrix. Different row or generator conventions multiply the Alexander polynomial of a knot and its state weights by the corresponding overall unit.