Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 112 / 1

Let $K_1$ be the connected sum of $2022$ trefoils. The trefoil has Seifert genus one, and additivity of Seifert genus gives

Take the twist knot whose standard diagram has $2020$ half-twists in its twist region and two crossings in its clasp. Every nontrivial twist knot has Seifert genus one. This diagram is a reduced alternating knot diagram, so the Tait crossing-number theorem says that its $2020+2=2022$ crossings realize the crossing number of a knot. Thus this knot has $g_s(K_2)=1$ and $c(K_2)=2022$.

Consider the symmetric Laurent polynomialIt satisfies $f(1)=1$, so the Alexander polynomial realization theorem gives a knot $K_3$ with ${\Delta }_{K_3}(t)\doteq f(t)$. The polynomial is not a unit of $\mathbb{Z}[t^{\pm 1}]$, whereas

Take the untwisted Whitehead double of a trefoil knot. The Whitehead pattern has winding number of a satellite pattern zero and becomes the unknot when its companion is the unknot. The Satellite formula for the Alexander polynomial therefore givesThe Whitehead pattern is geometrically essential in its solid torus, so the satellite knot with nontrivial companion $T_{2,3}$ is nontrivial. Hence $K_4$ is not isotopic to the unknot despite having trivial Alexander polynomial of a knot.