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

Abstract

For any given integer $r\ge 1$ and a quasitoric braid ${\beta }_r=({\sigma }_r^{-\epsilon }{\sigma }_{r-1}^{\epsilon }...$ ${\sigma }_1^{(-1{)}^r\epsilon }{)}^3$ with $\epsilon =\pm 1$, we prove that the maximum degree in $z$ of the HOMFLYPT polynomial $P_{W_2({\hat{\beta }}_r)}(v,z)$ of the doubled link $W_2({\hat{\beta }}_r)$ of the closure ${\hat{\beta }}_r$ 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.

33 pages, 27 figures