Gordian adjacency of T(4,5) and T(5,6) to T(2,7)

Determine whether the torus knots T(4,5) and T(5,6) are Gordian adjacent to the torus knot T(2,7), i.e., whether T(2,7) appears in a minimal unknotting sequence for T(4,5) and for T(5,6).

Background

Gordian adjacency K1 ≼ K2 means that K2 has a minimal unknotting sequence containing K1. Feller showed that T(2,7) is Gordian adjacent to many torus knots, providing a mechanism to construct connected sums that fail additivity of unknotting number.

The paper notes that while T(2,7) is known to be Gordian adjacent to all torus knots except possibly a short list, two cases—T(4,5) and T(5,6)—remain unsettled. Resolving these would complete the picture for small-parameter torus knots in this context.