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

For any given integer $r\ge 1$ and a quasitoric braid ${\beta }_r=({\sigma }_r^{-ϵ}{\sigma }_{r-1}^{ϵ}\mathrm{.}\mathrm{.}\mathrm{.}$ ${\sigma }_1^{(-1{)}^rϵ}{)}^3$ with $ϵ=\pm 1$, we prove that the maximum degree in $z$ of the HOMFLYPT polynomial $P_{W_2({\widehat{\beta }}_r)}(v,z)$ of the doubled link $W_2({\widehat{\beta }}_r)$ of the closure ${\widehat{\beta }}_r$ is equal to $6r-1$. As an application, we give a family ${\mathcal{K}}^3$ 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 ${\mathcal{K}}^3$ coincides with the canonical genus of its Whitehead double. Consequently, we give a new family ${\mathcal{K}}^3$ of alternating knots for which Tripp’s conjecture holds.