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

Induct on the Seifert genus $g_s(K)$. A genus-zero knot is the unknot. If $K$ is prime there is nothing to prove. Otherwise write $K=K_1\#K_2$ with both summands nontrivial. Part c givesso each summand has strictly smaller genus than $K$. Apply the induction hypothesis to both. Since the genus drops at every nontrivial split, the process terminates after finitely many steps and expresses $K$ as a finite connected sum of prime knots.

Solved by gpt-5.6-sol high.