The knot-complement theorem — Examples and Counterexamples

The matrix $A$ is integral with determinant $1$, so its linear map on ${\mathbb{R}}^2\mathrm{/}{\mathbb{Z}}^2$ is a torus homeomorphism. Its first column is $(1,0)$, hence it sends $\mu$ to $\mu$, and its second column is $(2,1)$, hence it sends $\lambda$ to $2\mu +\lambda$.

Define $F:V\to V$ by $F(z,w)=(w^{2}z,w)$. Since $\mathrm{\mid }w\mathrm{\mid }=1$, this maps $D^2\times S^1$ to itself, and its inverse is $(z^{\mathrm{\prime }},w^{\mathrm{\prime }})\mapsto ((w^{\mathrm{\prime }}{)}^{-2}{z}^{\mathrm{\prime }},w^{\mathrm{\prime }})$. Thus $F$ is a homeomorphism. On the meridian it has degree $1$; on the longitude it winds twice in the disk angle and once in the circle coordinate, giving the same matrix $A$.

The disk rotation $z\mapsto w^{2}z$ varies continuously with $w$ and contributes no orientation reversal; the circle coordinate is unchanged. Hence $F$ preserves the product orientation and is the required extension. This also instantiates [F1].

The crossing-free unknot diagram has one under-arc and no crossings, so [F1] gives a one-generator presentation with no relators and hence ${\pi }_1(S^3\setminus U,u_0)\cong \mathbb{Z}$ for any chosen complement basepoint $u_0$. Let $x_0$ be the trefoil basepoint in [F2]. If $U$ and $T$ were ambient equivalent, let $H$ be the final diffeomorphism of that ambient isotopy and choose $u_0=H^{-1}(x_0)$. By [F4] and [F5], $H$ would induce an isomorphism ${\pi }_1(S^3\setminus U,u_0)\cong {\pi }_1(S^3\setminus T,x_0)$. This contradicts [F2], so the knots are not ambient equivalent.

By [A1] and [L1], the countable-choice hypothesis of [F3] holds for both knots. Each abelianized peripheral system has $H_1\cong \mathbb{Z}$ with the meridian as generator and preferred longitude zero. The isomorphism sending $[{\mu }_U]$ to $[{\mu }_T]$ therefore sends the complete decorated triple $(H_1,[\mu ],[\lambda ])$ for $U$ to that for $T$. The systems agree although the knots do not, refuting the assertion.