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

Abstract

For any given integer r⩾1 and a quasitoric braid βr=(σr−ϵσr−1ϵ⋯σ1(−1)rϵ)3 with ϵ=±1, we prove that the maximum degree in z of the HOMFLYPT polynomial PW2(βˆr)(v,z) of the doubled link W2(βˆr) of the closure βˆr is equal to 6r−1. As an application, we give a family K3 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 K3 coincides with the canonical genus of its Whitehead double. Consequently, we give a new family K3 of alternating knots for which Trippʼs conjecture holds.