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

TL;DR

This paper establishes a precise relationship between the maximum degree of the HOMFLYPT polynomial and the canonical genus for a family of Whitehead doubles of alternating knots, confirming Tripp's conjecture for this family.

Contribution

It introduces a new family of alternating knots where the minimal crossing number equals the Whitehead double's canonical genus, validating Tripp's conjecture in this context.

Findings

Maximum degree in z of HOMFLYPT polynomial is 6r-1 for the family.

Minimal crossing number equals canonical genus for the family.

Tripp's conjecture holds for the newly identified family.

Abstract

For any given integer $r \geq 1$ 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.