For
$$
r_1=ad^{-1}b,
\qquad r_2=bd^{-1}c,
\qquad r_3=cd^{-1}a,
$$
the <Fox calculus> product rule and $\partial d^{-1}/\partial d=-d^{-1}$ give, in generator order $(a,b,c,d)$,
$$
\left(\phi\frac{\partial r_i}{\partial x_j}\right)
=
\begin{pmatrix}
1&t^{-1}&0&-t^{-1}\\
0&1&t^{-1}&-t^{-1}\\
t^{-1}&0&1&-t^{-1}
\end{pmatrix}.
$$
Multiplication of every row by the unit $t$ gives the equivalent <Alexander matrix>
$$
\boxed{A=
\begin{pmatrix}
t&1&0&-1\\
0&t&1&-1\\
1&0&t&-1
\end{pmatrix}.}
$$
For example, deleting the $a$-column gives determinant $-(t^2-t+1)$, the <Alexander polynomial of a knot> of the trefoil up to a unit.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-141/1/d/solution