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 knot diagram>[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 $c(K_2)=2022$.

Solution

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

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A twist knot is obtained by closing two strands with a clasp after inserting a row of half-twists. Every nontrivial twist knot has <Seifert genus> one; a standard reduced alternating diagram with $m$ crossings in the twist region has $m+2$ crossings.

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