Bridge number and unknotting number of an oriented knot

Assume the Axiom of Countable Choice. Identify the oriented $S^3$ with the unit sphere in ${\mathbb{R}}^4$ by an orientation-preserving diffeomorphism and fix the standard height $h(x_1,x_2,x_3,x_4)=x_4$. Its only critical points are the two poles, and each regular level is a 2-sphere. A Morse representative of a smooth oriented knot $K$ is a knot ambient isotopic to $K$ on which $h$ has only nondegenerate critical points. Define the bridge number $b(K)$ as the minimum number of local maxima of $h$ on a Morse representative of $K$. The standard unknot is the oriented unit circle in a fixed coordinate 2-plane of $S^3$.

A regular oriented diagram is a projection of a smooth oriented knot with finitely many transverse double points, no other multiple points, and an over/under choice at each double point. A crossing change switches which branch is over at one chosen double point and leaves the projection and strand orientations unchanged. Define the unknotting number $u(K)$ as the minimum number of crossing changes needed to turn some regular diagram of a representative of $K$ into a diagram of the standard unknot. Both minima are taken in subsets of $\mathbb{N}$. The Morse-representative set is nonempty by the first well-definedness result below; under countable choice the regular-diagram existence result and the descending-diagram theorem below make the unknotting set nonempty. The bridge-number definition and the least-element argument in $\mathbb{N}$ require no choice; countable choice is used only to invoke the stated regular-projection supplier for $u(K)$.

Smooth oriented knots and ambient isotopy in the standard oriented three-sphere

Finite transverse double points with strand orientations and over/under data

The standard unknot convention

The countable-choice axiom used for regular diagram existence