The canonical genus for Whitehead doubles of a family of alternating knots

For any given integer r ⩾ 1 and a quasitoric braid β r = ( σ r − ϵ σ r − 1 ϵ ⋯ σ 1 ( − 1 ) r ϵ ) 3 with ϵ = ± 1 , we prove that the maximum degree in z of the HOMFLYPT polynomial P W 2 ( β ˆ r ) ( v , z ) of the doubled link W 2 ( β ˆ r ) of the closure β ˆ r is equal to 6 r − 1 . As an application, we give a family K 3 of alternating knots, including ( 2 , n ) -torus knots, 2-bridge knots and alternating pretzel knots as its subfamilies, such that the minimal crossing number of any alternating knot in K 3 coincides with the canonical genus of its Whitehead double. Consequently, we give a new family K 3 of alternating knots for which Trippʼs conjecture holds.