Extension of Theorem 1.1 (prime outcomes from inter-component unknotting crossings) to allow 2-bridge summands

Prove that Theorem 1.1 (which constructs a diagram of K1 # K2 and a set C of u(K1) + u(K2) unknotting crossings such that changing any crossing in C yields a prime knot) holds without assuming that the prime summands K1 and K2 are not 2-bridge; that is, establish the same conclusion for arbitrary prime knots K1 and K2 subject to the theorem’s crossing-change hypotheses.

Background

Theorem 1.1 (stated earlier in the paper) shows that under mild assumptions on K1 and K2 (including that they are not 2-bridge), one can construct diagrams of K1 # K2 with an unknotting set of size u(K1)+u(K2) such that any single crossing change from this set produces a prime knot. This result provides counterexamples to a stronger additivity conjecture involving crossing arcs.

The authors suspect the non–2-bridge restriction is unnecessary and explicitly conjecture the theorem remains valid when K1 and K2 are allowed to be 2-bridge knots, motivating a sharpened understanding of prime tangles and their gluing behavior.