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

Forthe Fox calculus product rule and $\partial d^{-1}/\partial d=-d^{-1}$ give, in generator order $(a,b,c,d)$,Multiplication of every row by the unit $t$ gives the equivalent Alexander matrixFor example, deleting the $a$-column gives determinant $-(t^2-t+1)$, the Alexander polynomial of a knot of the trefoil up to a unit.