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

Abstract

For any given integer $r \geq 1$ and a quasitoric braid $β_r=(σ_r^{-ε} σ_{r-1}^ε...$ $ σ_{1}^{(-1)^{r}ε})^3$ with $ε=\pm 1$, we prove that the maximum degree in $z$ of the HOMFLYPT polynomial $P_{W_2(\hatβ_r)}(v,z)$ of the doubled link $W_2(\hatβ_r)$ of the closure $\hatβ_r$ 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.