Talk:Dual Thurston polytope of the 3-torus (Ex)

Let $\phi\in H^1(T^3;\mathbb{Z})$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\phi\in H^1(T^3;\mathbb{Z})$. Fix a basis $(x,y,z)$$(x,y,z)$ of $H_1(T^3;\mathbb{Z})$$H_1(T^3;\mathbb{Z})$. View $\phi$$\phi$ as a map from $\mathbb{Z}^3$$\mathbb{Z}^3$ to $\mathbb{Z}$$\mathbb{Z}$. Then the kernel of $\phi$$\phi$ is (at least) rank $2$$2$; rechoose the basis $(x,y,z)$$(x,y,z)$ so that $\phi(x)=\phi(y)=0$$\phi(x)=\phi(y)=0$. Then in $(x,y,z)$$(x,y,z)$-coordinates, we have $T^3=S^1\times S^1\times S^1$$T^3=S^1\times S^1\times S^1$, where $\phi(z)\cdot[S^1\times S^1\times 0]$$\phi(z)\cdot[S^1\times S^1\times 0]$ is the Poincaré dual of $\phi$$\phi$. Thus, the dual to $\phi$$\phi$ is represented by a disjoint union of $|\phi(z)|$$|\phi(z)|$ tori, so $x_{T^3}(\phi)=0$$x_{T^3}(\phi)=0$.

Let $v\in H_1(T^3;\Z)$$v\in H_1(T^3;\Z)$. We have $v\in T(T^3)^*$$v\in T(T^3)^*$ if and only if $\phi(v)\le x_{T^3}(\phi)=0$$\phi(v)\le x_{T^3}(\phi)=0$ for all $\phi\in H^1(T^3;\mathbb{Z})$$\phi\in H^1(T^3;\mathbb{Z})$. Since $\phi(v)=-(-\phi)(v)$$\phi(v)=-(-\phi)(v)$, we have $v\in T(T^3)^*$$v\in T(T^3)^*$ if and only if $\phi(v)=0$$\phi(v)=0$ for all $\phi$$\phi$. Therefore, $v\in T(T^3)^*$$v\in T(T^3)^*$ if and only if $v=0$$v=0$.

Thus, the dual Thurston polytope $T(T^3)^*$$T(T^3)^*$ is the single point $(0,0,0)$$(0,0,0)$.