Coprime torus knot

Assume the Axiom of Countable Choice. Identify the standard oriented three-sphere with $S^3=\{(z_1,z_2)\in {\mathbb{C}}^2:\mathrm{\mid }{z}_1{\mathrm{\mid }}^2+\mathrm{\mid }{z}_2{\mathrm{\mid }}^2=1\}$ and choose $0<r<1$. The standard unknotted Heegaard torus is $T_r=\{\mathrm{\mid }{z}_1\mathrm{\mid }=r,\mathrm{\mid }{z}_2\mathrm{\mid }=\sqrt{1-r^2}\}$; its two sides are solid tori. In the angular coordinates $(u,v)\in (\mathbb{R}\mathrm{/}\mathbb{Z}{)}^2$ given by $(u,v)\mapsto (r{e}^{2\pi iu},\sqrt{1-r^2}{e}^{2\pi iv})$, let $\alpha$ and $\beta$ be the first and second coordinate classes in $H_1(T_r;\mathbb{Z})$. For coprime nonzero integers $p,q$, the oriented torus knot $T(p,q)$ is the image of $s\mapsto (r{e}^{2\pi ips},\sqrt{1-r^2}{e}^{2\pi iqs})$, for $s\in \mathbb{R}\mathrm{/}\mathbb{Z}$, oriented by increasing $s$; it represents the primitive slope $p\alpha +q\beta$. The mirror convention is the reflection $(z_1,z_2)\mapsto (z_1,\overline{z_2})$, which sends $T(p,q)$ to $T(p,-q)$ with image orientation, and reversing the knot orientation gives $T(-p,-q)$. For nonzero $p,q$ with $d=\gcd (\mathrm{\mid }p\mathrm{\mid },\mathrm{\mid }q\mathrm{\mid })>1$, the corresponding torus link is the union of $d$ disjoint parallel copies of the primitive slope $(p\mathrm{/}d,q\mathrm{/}d)$; it is not a single embedded curve of class $p\alpha +q\beta$. The exterior is $E_K=S^3\setminus \mathrm{int}N(K)$ for a chosen closed tubular neighborhood. Orient each normal disk so that a positive tangent vector to $K$ followed by a positive basis of the disk gives the orientation of $S^3$. A positive meridian is the boundary of such a disk with its induced boundary orientation. Let $\pi :\mathrm{\partial }{E}_K\to K$ be the projection of the chosen normal-disk tube. A preferred longitude is an oriented simple closed curve $\lambda \subset \mathrm{\partial }{E}_K$ for which $\pi {\mathrm{\mid }}_{\lambda }$ has degree $+1$ and $i_{\ast }[\lambda ]=0$ in $H_1(E_K;\mathbb{Z})$, where $i:\mathrm{\partial }{E}_K\hookrightarrow E_K$ is inclusion; this definition alone asserts neither existence nor uniqueness.

The chosen compact exterior convention

The positive normal-disk meridian and degree-one preferred-longitude convention

Cattabriga and Francaviglia, §5.1

Calegari, Example 1.2

The rectangle is a $2$-by-$3$ block of unit fundamental squares in the universal cover ${\mathbb{R}}^2$ of $T_r$. The segment from $(0,0)$ to $(2,3)$ projects to the slope $2\alpha +3\beta$; its endpoints differ by a lattice vector, so the projection closes on the torus. The picture records the slope only; the proof of embeddedness is the coprimality argument in the Definition.