Crossing number of an alternating knot and canonical genus of its Whitehead double

TL;DR

This paper investigates the relationship between the crossing number of alternating knots and the canonical genus of their Whitehead doubles, disproving a previous conjecture for certain 3-braid knots and proposing a new conjecture.

Contribution

It disproves Tripp's conjecture for specific alternating 3-braid knots and introduces a revised conjecture linking crossing number and Whitehead double genus for prime knots.

Findings

Disproved the conjecture for certain alternating 3-braid knots.

Identified a new class of prime alternating knots satisfying the conjecture.

Proposed a modified conjecture relating crossing number and Whitehead double genus.

Abstract

A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of alternating knots including $(2, n)$ torus knots, $2$-bridge knots, algebraic alternating knots, and alternating pretzel knots. In this paper, we prove that the conjecture is not true for any alternating $3$-braid knot which is the connected sum of two torus knots of type $(2, m)$ and $(2, n)$. This results in a new modified conjecture that the crossing number of any prime knot coincides with the canonical genus of its Whitehead double. We also give a new large class of prime alternating knots satisfying the conjecture, including all prime alternating $3$-braid knots.