Oriented knot connected sum is a commutative monoid

For commutativity, choose the symmetric standard two-ball model in [F1]. The orientation-preserving half-turn exchanging its two lobes carries each oriented knot tangle to the corresponding tangle in the swapped model and carries the fixed joining map to itself. It therefore gives an orientation-preserving ambient isotopy between the model outputs for $K\mathrm{\#}J$ and $J\mathrm{\#}K$; [F2] transfers this equality to arbitrary connected-sum data.

For associativity, place representatives of $K,J,L$ in three disjoint standard ball-arc pieces in a row and join consecutive pieces by the same fixed gluing. The constructions $(K\mathrm{\#}J)\mathrm{\#}L$ and $K\mathrm{\#}(J\mathrm{\#}L)$ then use the same three pieces and the same two joining maps; only the order in which the two gluings are performed differs. Since the gluings have disjoint boundary spheres, their order does not change the resulting oriented manifold or knot. By [F2], this common model represents both bracketings for arbitrary data.

For the identity, choose the standard unknot summand with a small ball meeting it in a trivial arc. Its spanning disk shows that the complementary arc is boundary-parallel and hence trivial. The connected-sum gluing replaces a short trivial arc of $K$ by another standard trivial arc. The ball-arc isotopy of [F2] identifies the output with $K$. Reversing the order gives the same argument for the left identity.