Tunnel number degeneration under the connected sum of prime knots

We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesnʼt degenerate by more than one: t ( K 1 # K 2 ) ⩾ t ( K 1 ) + t ( K 2 ) − 1 , for K 1 and K 2 prime knots.