Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 112 / 1 / c

Boundary-connected-summing minimal Seifert surfaces for $K$ and $K^{\prime }$ givesFor the reverse inequality, let $F$ be a minimal-genus Seifert surface for the connected sum of knots and let $S$ be its standard splitting sphere. A minimal-genus Seifert surface is incompressible in the knot exterior: a compression either lowers its genus or separates off a closed component that can be discarded. Put $F$ and $S$ in transverse position and minimize the number of intersection circles. An innermost circle on $S$ either gives a compression of $F$ or bounds a disk on $F$ across which it can be removed. Both alternatives contradict minimality, so $F\cap S$ consists only of the single arc joining the two points of $S\cap \partial F$.

Cutting $F$ along this arc gives Seifert surfaces $F_1,F_2$ for $K,K^{\prime }$. Their Euler characteristics satisfywhich, since all three surfaces have one boundary component, is equivalent toThis proves additivity.

The torus knot $T_{2,3}$ bounds a once-punctured torus, and its degree-two Alexander polynomial of a knot forces every Seifert surface to have genus at least one. Thus $g_s(T_{2,3})=1$. If it were a composite knot, both nontrivial summands would have positive Seifert genus, and additivity would give genus at least two. Hence $T_{2,3}$ is a prime knot.

Solved by gpt-5.6-sol high.