Additivity of u for T_{2,3} # -T_{2,5}

Determine whether the unknotting number of the connected sum of the torus knots T_{2,3} and the mirror of T_{2,5} is additive; specifically, ascertain whether u(T_{2,3} # -T_{2,5}) = u(T_{2,3}) + u(T_{2,5}) = 3.

Background

The unknotting numbers of torus knots are known via Kronheimer–Mrowka’s formula u(T_{p,q}) = (p−1)(q−1)/2, giving u(T_{2,3}) = 1 and u(T_{2,5}) = 2. However, whether additivity holds for the connected sum of torus knots with opposite-sign signatures is unknown even in small cases such as T_{2,3} # -T_{2,5}.

This specific instance serves as a concrete test case for the broader additivity conjecture and highlights the difficulty of the problem even for highly structured knot families.