Torus links as closures of two-strand braids

The literal word model and component count. Write $R=\sqrt{1-r^2}$. The finite signed half-turn word has disk strands $\pm r{e}^{i\mathrm{\Theta }(t)}$, where $\mathrm{\Theta }$ is smooth, $\mathrm{\Theta }(0)=0$, $\mathrm{\Theta }(1)=m\pi$ and all positive-order endpoint derivatives vanish. Take $\mathrm{\Theta }=0$ at $m=0$; concatenating the finitely many flattened half-turn angles gives such a $\mathrm{\Theta }$ for every other $m$. The two points stay distinct. Its literal smooth closure has $z=R{e}^{2\pi it}$, $w=\pm r{e}^{i\mathrm{\Theta }(t)}$ by [F1]. The endpoint permutation is $(12{)}^m$, so there is one component when $m$ is odd and two when $m$ is even, including $m=0$. This is $\gcd (2,m)$, with the positive gcd convention for negative $m$.

An explicit ambient isotopy to the uniform torus model. Set $\mathrm{\Delta }(t)=m\pi t-\mathrm{\Theta }(t)$. Its values at both ends are zero, its first derivatives at both ends are $m\pi$, and all higher endpoint derivatives agree; hence it is a smooth function on the page circle. Choose a fixed smooth radial cutoff $\chi (\mathrm{\mid }x{\mathrm{\mid }}^2)$ equal to one near $r^2$ and zero near $\mathrm{\mid }x\mathrm{\mid }=1$. On $V$ use $H_s([(x,t)])=[(e^{is\chi (\mathrm{\mid }x{\mathrm{\mid }}^2)\mathrm{\Delta }(t)}x,t)]$. Its inverse rotates by the negative angle, since $\mathrm{\mid }x\mathrm{\mid }$ is unchanged. It is a smooth isotopy, and through the explicit $\phi$ it extends by identity across the axis because the cutoff is zero near the disk boundary. It preserves orientation as a smooth path of diffeomorphisms starting at identity. At $s=1$ the curve is exactly $z=R{e}^{2\pi it}$, $w=\pm r{e}^{im\pi t}$. For odd $m$, concatenate its two strands with parameter $u\in \mathbb{R}\mathrm{/}2\mathbb{Z}$; this gives $z=R{e}^{2\pi iu}$, $w=r{e}^{im\pi u}$, with winding pair $(2,m)$ after parameter $u\mathrm{/}2$. For even $m$, each strand is a circle with winding pair $(1,m\mathrm{/}2)$ and the two are disjoint. These are the standard torus curves $T(2,m)$, including the stated component count. No generic continuous-braid smoothing theorem or Markov-preservation assumption is used.

The two one-crossing oriented unknots without Choice. For $m=ϵ=\pm 1$, use the chart of $S^3\setminus \{w=0\}$ with $\theta =\arg w$ and $\xi =z{e}^{-2ϵi\theta }$. Its inverse is $(\xi ,\theta )\mapsto (\xi e^{2ϵi\theta },\sqrt{1-\mathrm{\mid }\xi {\mathrm{\mid }}^2}{e}^{i\theta })$, so it is explicitly $D_{\xi }^{\circ }\times S^1$. The curve from step 2.1 has $\xi =R$ constant. Move this disk point along the real segment from $R$ to $0$ by finitely many small cutoff translations $F_s(x)=x+s\eta (x)v$, with compact support in $D_{\xi }^{\circ }$ and $\mathrm{\parallel }D\eta {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }v\mathrm{\mid }<1$. A fixed smooth bump and a sufficiently fine finite subdivision suffice because that compact segment has positive distance $1-R$ from the boundary. The inverse is the unique limit of $x_{k+1}=y-s\eta (x_k)v$, starting at $x_0=y$: successive differences are bounded by a geometric sequence with ratio at most $\mathrm{\parallel }D\eta {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }v\mathrm{\mid }<1$, so the explicit sequence converges and its fixed point is unique. The Jacobian has determinant $1+sD\eta \cdot v>0$; [F3] makes the already constructed inverse smooth locally and hence globally. Each map is identity outside its compact interior support, so gives a disk diffeomorphism. Compose these finitely many disk-map families with the same parameter $s$; the finite smooth composition starts at identity and at $s=1$ sends $R$ to $0$. Its product with the unchanged $\theta$ extends by identity across $\{w=0\}$, since $\mathrm{\mid }\xi \mathrm{\mid }\to 1$ there. This ambient isotopy takes the curve to $(0,e^{i\theta })$. The explicit unitary rotation $(z,w)\mapsto (\cos az+\sin aw,-\sin az+\cos aw)$, $0\le a\le \pi \mathrm{/}2$, then takes it to the trivial one-braid circle $(e^{i\theta },0)$. For $ϵ=1$ its orientation is already the positive page orientation. For $ϵ=-1$ reverse that circle's negative angular orientation using $(z,w)\mapsto (\bar{z},\bar{w})$; this map is the endpoint of the explicit $\pi$ rotation in the real plane of the two imaginary coordinates, an orientation-preserving path in $SO(4)$. Thus both signs give the oriented unknot. Every construction is explicit or a finite selection; no countable choice is used.

The unlink, trefoils and conclusion. At $m=0$ step 1.1 is the literal trivial two-braid closure, whose explicit disjoint spanning disks are [F1]. At $m=\pm 3$, step 2.1 gives the standard $(2,\pm 3)$ torus knots, the trefoils named in the Statement. The map $(z,w)\mapsto (z,\bar{w})$ changes $m$ to $-m$ and reverses the ambient orientation, so the two are mirrors. Steps 1.1-3.1 give the component formula and both one-crossing unknots, and the coordinate curves verify $T(2,m)$ for every integer $m$.