Thurston unit ball of a family of $n$-chained links and their fibered face

Let $M$ be a $3$-dimensional manifold. We will suppose that $M$ has tori boundaries for simplicity. In one of his many seminal works [20], W. Thurston introduced a notion of a semi-norm for the second homology vector spaces of $M$. More precisely, let $[a]\in H_{2}(M,\partial M;\mathbb{Z})$ be an integral second homology class. Then $[a]$ can be represented by a disjoint union of properly embedded surfaces $S_{i}$. The Thurston norm of $[a]$ is then defined to be

where the minimum is taken over all possible ways to represent $[a]$ as a disjoint union of properly embedded surfaces. If $M$ is irreducible and atoroidal, this then extends to a norm on $H_{2}(M,\mathbb{R})$. We sometimes use $||\cdot||$ to denote the Thurston norm. In the same paper, he proves that the unit ball with respect to that norm, that we will call Thurston unit ball, is always a polytope. Even though this concept has had huge theoretical consequences, it seems that there are very few cases for which we know precisely what the Thurston unit ball is. An interesting question in that regard is the following.

This question was already posed by Kitayama in [13]. It has been generalized in terms of groups and their $1$st homology by Friedl, Lück and Tillmann [5]. In [17], Pacheco-Tallaj, Schreve and Vlamis find out the shape of the Thurston unit ball for tunnel number-one manifolds. We refer [13] for more recent research of Thurston norm.

In this article we show that the Thurston unit ball can contain highly symmetric polytopes in arbitrary high dimensions. We will be interested in determining the Thurston unit ball for a family of complements of links, denoted by $C(n,p)$. Briefly speaking, $C(n,p)$ is an $n$-chained link with $p$ positive half-twist on the first component if $p$ is positive or $p$ negative half-twist on the first component if $p$ is negative (see figure 4). The complements of these links are in some sense generalizations of the magic manifold, which is the complement of $C(3,0)$. The magic manifold and its properties are thus good examples to keep in mind.

In a previous article [2], the two authors together with Harry Baik and Changsub Kim studied the relation between the minimal entropy of pseudo-anosov maps on a surface $S$ and the action of these maps on $H_{1}(S)$. In order to do so, the use of the complement of $C(n,-2)$ was crucial.

Here are the main results of this article.

Moreover, we also find a fibered face in each case, and determine the topological type of every fibered surface in that fibered face.

A complete answer for the case of $p<0$ is out of our reach for now. We nonetheless find a set $V(n,p)$ of points in the Thurston unit ball and conjecture that their convex hull, denoted by $B(n,p)$ is the whole Thurston unit ball. This conjecture is partially supported by computational data, obtained using the program Tnorm and gathered in Appendix B.

For the case of $p=-2$, we also managed to compute the Teichmuller polynomial for every value of $n$.

We gather here the essential tools that will be used in the rest of the paper.

In [11], Eiko Kin analysed in detail a $3$-manifold, known as the magic manifold. This manifold has the property that all the faces of its Thurston unit ball are fibered. She was able to precisely determine all the fibered faces and, for each integer point in a fibered face, find the topology of the associated monodromy (i.e determine its genus and the number of boundary components). In this section we generalize the technique used for the magic $3$-manifold to study sequences of fibers in more general link complements. We investigate whether these $n$-chained links are fibered, what are the fibers and the associated monodromies, and the shape of Thurston unit norm ball of various $C(n,p)$’s.

Let $M(n,p)$ be the complement of a small enough neighborhood $\mathcal{N}(C(n,p))$ of $C(n,p)$. For example, $M(3,0)$ is the magic 3-manifold. Note that $\partial\mathcal{N}(C(n,p))$ is a disjoint union of $n$ tori. In [16], Neumann and Reid prove that $M(n,p)$ with $n\geq 3$ is hyperbolic if and only if $\{|n+p|,|p|\}\not\subseteq\{0,1,2\}$. Moreover it is fibered, which is proven by Leininger [14]. More precisely, Leininger shows that, except for $(n,p)=(2,-1)$, $M(n,p)$ is fibered if $n\geq-p\geq 0$ by providing an actual fibers. We will use that construction later.

Now, we focus on the homology of $M(n,p)$. Consider that we draw $C(n,p)$ in such a way that the top link has the $p$ half-twists, as in the figure 4. We then denoted the top link component as $L_{1}$, and we enumerate the other components $L_{2},L_{3},\cdots L_{n}$ in a clockwise fashion.

Each $L_{i}$ bounds a twice punctured disk, the punctures coming from $L_{i-1}$ and $L_{i+1}$. We will denote this twice punctured disk by $K_{i}$. Note that $\{[K_{i}]_{1\leq i\leq n}\}$ forms a basis of $H_{2}(M,\partial M)$. Then we have

Note that the fiber $S$ is a genus $1$ surface with $n$ boundaries and so its Euler characteristic is equal to $n$. Our next goal is to find the fibered face $\mathcal{F}$ that contains $S$.

Since the link $L=C(n,p)$ with $p>0$ is admits an alternating link diagram as we defined, its Seifert surface $S$ is the minimal genus surface for $L$.

We describe a simple algorithm to compute the genus of $C(n,p)$ using the Seifert algorithm.

The above lemma implies that the Thurston unit ball of $M\cong M(n,p)$ is an $n$ dimensional cocube.

In [14] Leininger proved that $C(n,-p)$ is fibered for $0\leq p\leq n$. In this section we investigate what is the shape of the Thurston unit ball for the complements of $n$-chained links with negative twists. Suppose we have a $n$-chain link $C(n,-p)$ and that we have labeled each link components as before. Note that we can untwist all the negative twists. After resolving the negative twists on $L_{1}$, our link became a chain link with no twists. However the shape of clasps may have changed during this process. We re-assign the orientations of each $L_{i}$’s in a circular way after resolving all the twists on $L_{1}$, and, for each $i=1,\cdots,n$, we define $e_{i}\in H_{2}(M(c,p),\partial M(c,p);\mathbb{Z})$ to be the twice punctured disk bounded and oriented by $L_{i}$. See the figure 8.

Each link component has $2$ clasps, which may now be $+$ clasp or $-$ clasp. Since the $-$ clasp only arise whenever one negative twist is resolved, the number of $-$ clasps is equal to $|p|$ after resolving all twists. We define the shape vector of $C(n,p)$.

Note that the number of $-$ entries in any shape vector is equal to $|p|$.

Suppose $L_{i}$ has $-$ shape with $L_{i-1}$ and $+$ shape with $L_{i+1}$. Here are $2$ isotopic operations we can perform on each $L_{i}$.

1. (1)

   A flip: we flip $L_{i}$ so that the $+$ clasp changes to a $-$ clasp and vice versa. Hence a flip exchanges the $(i-1)$’th entry and $i$’th entry of the shape vector.

2. (2)

   A full twists: There is a twice punctured disc $D$ bounded by $L_{i}$ whose second homology class is $e_{i}$. Cut $M(n,p)$ at $D$. At the slice, there are $2$ copies of $D$, say $D_{1},D_{2}$. Now re-glue $D_{1}$ and $D_{2}$ after twisting either $D_{1}$ or $D_{2}$ by $360$ degrees.

Now we have the following proposition.

Start with $C(n,-1)$. After untwisting once, we obtain an $n$-chained link with shape vector $(-,+,\cdots,+)$. By proposition 6.2, $\frac{1}{n-3}(1,0,1,\cdots,1)$ is a point of Thurston norm equal to $1$. If we flip $L_{2}$, then the shape vector changes to $(+,-,+,\cdots,+)$. Now we can perform a full twists on $L_{3}$ and then, using the same method as in the proof of proposition 6.2, we can deduce that the point $(1,-1,0,1,\cdots,1)$ also is on the unit sphere. Note that we have a $-1$ on the second entry this time. Simply put, as the $0$ coordinate moves one step on the right, it also introduces a minus sign. Hence, by this process, we get a total of $2n$ points on the unit sphere, namely

and their antipodal points.

Now consider $C(n,-2)$. The shape vector contains two $-$ entries. We can perform a full twist on the two link components $L_{i}$ and $L_{j}$ whose clasps on both sides are different, unless $L_{i}$ and $L_{j}$ are consecutive link components. In this case, we get $2$ zero entries in the new points and hence it represents a sphere with $n-2$ punctures. Therefore it has Thusrton norm $n-4$. See the figure 8 and 9 also.

Therefore, we get the following corollary.

We describe the topological property of various $B(n,p)$’s.

Some faces of the polytope $B$ are actually faces of the Thurston unit ball. We introduce another isotopic operation of for link components which have the same clasp shape on both sides.

The theorem implies that most of the faces in $B(n,p)$ are actually fibered faces of the Thurston unit norm ball. We end this section with a question and some remarks.

As far as we know, the answer is positive. We thanks William Worden and the program Tnorm[21] which helped us to calculate and verify the theorems. We provide the table of all the vertices of the Thurson unit normal ball , calculated by Tnorm, for various $C(n,p)$’s up to $n\leq 6$ in the appendix B.

As we already mentioned, $C(n,p)$ is fibered for $0\leq-p\leq n$. Assigning proper signs, Leininger’s fibered surface has coordinates $(1,\cdots,1)$ and has genus $1$ and $n$ punctures. Since each vector $e_{i}$ of the canonical basis represents a twice punctured disk, we can deduce that there is a fibered face $\mathcal{F}$ which contains the standard $(n-1)$ simplex spanned by the $\{e_{i}\}$’s. Furthermore, using lemma 4.2 and similar methods as in the proof of corollary 4.3, we can get that the Euler characteristic of any primitive points of $(x_{1},\cdots,x_{n})$ with all positive entries is equal to $\sum_{i=1}^{n}x_{i}$. We provide some calculations for the $p<0$ case in the appendix B. In the remaining section, we will cover the special case of $C(n,-2)$, in which case more explicit calculations can be made.

In this section we compute explicitly the Teichmüller polynomials for one fibered face of $C(n,-2)$. Let $M_{n}$ be the exterior complement of the link $C(n,-2)$. We denote the surface obtained from performing the Seifert algorithm to $C(n,-2)$ by $S_{n}$. We will sometimes omit the subscript $n$ if it is not important in the context. Since $M_{n}$ is the complement of $C(n,-2)$, the second homology $H_{2}=H_{2}(M_{n},\partial M_{n};\mathbb{Z})$ is a free abelian group of rank $n$, with a canonical basis given by the meridians of the link components. With that in mind, we remark that $S_{n}$ is a surface of genus one with $n$ boundaries and its coordinates in $H_{2}$ are $(1,1,\cdots,1)$. Since $S_{n}$ is a Murasugi sum of one horizontal Hopf bands with $n$ vertical Hopf bands, it is a fiber. By Gabai’s theorem, the monodromy $\varphi_{n}$ of this fibering is the composition of the Dehn twists along the cores of the Hopf bands.

Thus, if $S_{n}$ is placed as suggested in figure 12, the monodromy $\varphi_{n}$ is the composition of the $n$ vertical multi-twists directed downward followed by the left Dehn twist along the core of horizontal band. Following the methods in [3] and [15], we now compute the Teichmüller polynomial corresponding to the fibered face of the Thurston unit ball which contains the fiber $(1,\cdots,1)$.

As explained in section 2.3, we first need to compute $H=\operatorname{Hom}{(H^{1}(S,\mathbb{Z})^{\varphi},\mathbb{Z})}$ and then understand how the lift $\tilde{\varphi}_{n}$ of $\varphi_{n}$ acts on the cover $\tilde{S}_{n}$ of $S_{n}$ which has $H$ as a deck transform group. In this case, as noted in [3], the group $H$ is simply equal to the $\varphi_{n}$ invariant first homology $H_{1}(S_{n}:\mathbb{Z})^{\varphi_{n}}$ . We choose $c_{0},\cdots,c_{n}$ as a basis for $H_{1}(S_{n};\mathbb{Z})$, where $c_{0}$ is the curve corresponding to the core of the horizontal band and $c_{1},\cdots,c_{n}$ are the curves corresponding to the cores of the vertical bands, $c_{1}$ being the leftmost one and $c_{n}$ the rightmost one. Then $H_{1}(S_{n}:\mathbb{Z})^{\varphi_{n}}$ is the subspace of $H_{1}(S_{n};\mathbb{Z})$ generated by the column vectors of

We now need to figure out what the cover $\tilde{S}_{n}$ is and how $\tilde{\varphi}_{n}$ acts on it. Once again, the details are all given in [3], so instead of repeating them here, we will give some graphical explanation for the simplest no trivial example, that is $M_{3}=C(3,-2)$. In this case, the cover $\tilde{S}_{n}$ is explicitly drawn in figure 14.

Let $T$ be the matrix representing the $H$-module action of $\tilde{\varphi}_{n}$ on $\tilde{S}_{n}$. The monodromy $\varphi_{n}$ being the composition of one horizontal Dehn twist and $n$ horizontal ones, we can decompose the matrix $T$ into $T_{V}$ and $T_{H}$. These matrices represent the action of the lifts of the vertical multi-twist and the horizontal Dehn twist on $\tilde{S}$. Note that the entries of these matrices are in $\mathbb{Z}[G]$, where $G$ is a deck transformation of $\widetilde{S}$, isomorphic to $\mathbb{Z}^{n-1}$.

Using our conventions, $T_{V},T_{H}$ are $2n\times 2n$ matrices whose entries are as below;