Diffeomorphism of the double branched cover Σ₂(τ⁰ρ¹(J)) with S⁴

Determine whether the two-fold branched cover Σ₂(τ⁰ρ¹(J)) of S⁴, branched along the 2-knot that is the 0-twist 1-roll spin of the pretzel knot J = P(−2,3,7), is diffeomorphic to the standard 4-sphere S⁴.

Background

The authors show strong stabilization results for double branched covers of twist-roll spun knots and for connected sums with projective planes, trivializing certain homotopy complex projective planes. However, despite these advances, they cannot conclude that the associated homotopy 4-spheres are standard.

In particular, for the pretzel knot J = P(−2,3,7), they ask whether the specific double branched cover Σ₂(τ⁰ρ¹(J)) is diffeomorphic to S⁴, which would settle the smooth standardness question for this case. This question is framed as a subquestion related to Miyazawa’s work.