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

摘要

For any given integer r ≥ 1 and a quasitoric braid β_r = (σ_r~(-∈)σ_(r-1)~∈ …σ_1~((-1)~r∈)~3 with ∈ = ±1, we prove that the maximum degree in z of the HOMFLYPT polynomial P_(W2(βr))(v.z) of the doubled link W_2(βr) of the closure βr is equal to 6r - 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 Tripp's conjecture holds.