Algorithms and complexity for Turaev–Viro invariants

Abstract

The Turaev–Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. The invariants are parameterized by an integer $r \ge 3$. We resolve the question of complexity for $r=3$ and $r=4$, giving simple proofs that the Turaev–Viro invariants for $r=3$ can be computed in polynomial time, but computing the invariant for $r=4$ is #P-hard. Moreover, we describe an algorithm for arbitrary r, which is fixed-parameter tractable with respect to the treewidth of the dual graph of the input triangulation. We show through concrete implementation and experimentation that this algorithm is practical—and indeed preferable—to the prior state of the art for real computation. The algorithm generalises to every triangulated 3-manifold invariant defined from tensor network contraction.

1 Introduction

In geometric topology, testing homeomorphism (topological equivalence) is a fundamental algorithmic problem. However, beyond dimension two it is remarkably difficult. In dimension three—the focus of this paper—an algorithm follows from Perelman’s proof of the geometrization conjecture (Kleiner and Lott 2008), but it is extremely intricate, its complexity is unknown and it has never been implemented.

As a result, practitioners in computational topology rely on simpler topological invariants—computable properties of a topological space that can be used to tell different spaces apart. One of the best known invariants is homology, but for 3-manifolds (the 3-dimensional generalisation of surfaces) this is weak: there are many topologically different 3-manifolds that homology cannot distinguish. Therefore major software packages in 3-manifold topology rely on invariants that are stronger but more difficult to compute.

In the discrete setting, there exist large numbers of invariants for 3-manifolds defined with the help of triangulations and state sums, i.e., large sums of weights associated to colourings of the edges of a triangulation (Barrett and Westbury 1996; Turaev 2010). Among the most commonly used such state sum invariants are the Turaev–Viro invariants (Turaev and Viro 1992).

They can be computed using the major software packages Regina (Burton et al. 1999) and Manifold Recogniser (Matveev 2003; Matveev et al. 2012), and they play a key role in developing census databases, which are analogous to the well-known dictionaries of knots (Burton 2007; Matveev 2003). However, current implementations (Burton et al. 1999; Matveev et al. 2012) are based on backtracking searches, and require exponential time.¹

The purpose of this paper is threefold: (1) to introduce the Turaev–Viro invariants (and their close relatives) to the wider computational topology community; (2) to understand the complexity of computing these invariants; and (3) to develop new algorithms that are suitable for practical software.

The Turaev–Viro invariants are parameterized by an integer r, $r \ge 3$, and additionally depend on a 2r-th root of unity $\zeta = e^{i \pi s / r}$, $0< s < 2r$ coprime to r; we denote these invariants by $\mathrm {TV}_{r}$. A typical algorithm for computing $\mathrm {TV}_{r}$ takes as input a triangulated 3-manifold, composed of n tetrahedra attached along their triangular faces; we use n to indicate the input size. For all known algorithms, the difficulty of computing $\mathrm {TV}_{r}$ grows significantly as r increases (but in contrast, the difficulty is essentially independent of $\zeta$).

Our main results are as follows.

• Kauffman and Lins (1991) state that for $r=3,4$ one can compute $\mathrm {TV}_{r}$ via “simple and efficient methods of linear algebra”, but they give no details on either the algorithms or the complexity. We point out that in fact the situations for $r=3$ and $r=4$ are markedly different: computing $\mathrm {TV}_{r}$ for orientable manifolds and $r=3$ can be done in polynomial time, but for $r=4$ it is #P-hard.

• We give an explicit algorithm for computing $\mathrm {TV}_{r}$ for general r that is fixed-parameter tractable (FPT) in the treewidth of the dual graph of the input triangulation. Specifically, for any fixed r and any class of input triangulations whose dual graphs have bounded treewidth, the algorithm has running time linear in n. Furthermore, we show through comprehensive experimentation that this algorithm is practical. That is, we implement it in the open-source software package Regina (Burton et al. 1999), run it through exhaustive census databases, and find that this new FPT algorithm is comparable to—and often significantly faster than—the prior backtracking algorithm.

• We sketch how this algorithm naturally generalises to all invariants which can be expressed as a state sum over tensor network contractions. In particular, we describe FPT-algorithms for all invariants arising from the very general Turaev–Viro–Barrett–Westbury construction for triangulated 3-manifolds (Barrett and Westbury 1996).

• We give a new geometric interpretation of the formula for $\mathrm {TV}_{r}$, based on systems of “normal arcs” in triangles. This generalises earlier observations of Kauffman and Lins for $r=3$ based on embedded surfaces (Kauffman and Lins 1991), and offers an interesting potential for future algorithms based on Hilbert bases.

Proving #P-hardness for $r=4$ and polynomial-time solvability for $r=3$ is simple: the algorithm for $r=3$ derives from a known homological formulation (Matveev 2003), and the result for $r=4$ is essentially Kirby and Melvin’s NP-hardness proof for the more complex Witten–Reshetikhin–Turaev invariants (Kirby and Melvin 2004). The latter result complements other hardness results for the Turaev–Viro invariants in the literature: It is known that approximating Turaev–Viro invariants for triangulated manifold is both #P-hard (Alagic and Lo 2017, Theorem 5.1) and universal for quantum computations (Alagic et al. 2010) for infinitely many values of r.

The FPT algorithm for general r is significant in that it is not just theoretical, but also practical—and indeed preferable—for real software. It was previously known that computing general tensor network contractions is FPT in the treewidth, see Markov (2008) where this result is used to simulate quantum computations efficiently. Moreover, a specialised FPT result for $\mathrm {TV}_{r}$ is explicitly mentioned in Burton and Downey (2014). However, these prior result are purely theoretical, and lead to infeasibly large constants in the running time if translated to a concrete algorithm. More generally, FPT algorithms do not always translate well into practical software tools, and this paper is significant in giving the first demonstrably practical FPT algorithm in 3-manifold topology. The FPT algorithm is implemented in the open-source software package Regina (Burton et al. 1999).

An extended abstract of this work has appeared in the proceedings of ICALP 2015 (Downey et al. 2015).

2 Preliminaries

Let M be a closed 3-manifold. A generalised triangulation of M is a collection of n abstract tetrahedra $\varDelta _1,\ldots ,\varDelta _n$ equipped with affine maps that identify (or “glue together”) their 4n triangular faces in pairs, so that the underlying topological space is homeomorphic to M.

In particular, as a consequence of the face identifications, it is possible that several vertices of the same tetrahedron may be identified together (and likewise for edges and triangles). Indeed, it is common in practical applications to have a one-vertex triangulation, in which all vertices of all tetrahedra are identified to a common point. In general, the 4n tetrahedron vertices are partitioned into equivalence classes according to how they are identified together; we refer to each such equivalence class as a single vertex of the triangulation, and likewise for edges and triangles.

Generalised triangulations are widely used across major 3-manifold software packages. They are (as the name suggests) more general than simplicial complexes, which allows them to express a rich variety of different 3-manifolds using very few tetrahedra. For instance, with just $n \le 11$ tetrahedra one can create 13,400 distinct prime orientable 3-manifolds (Burton 2011; Matveev 2003).

2.1 The Turaev–Viro invariants

Let $\mathfrak {T}$ be a generalised triangulation of a closed 3-manifold M, and let $r \ge 3$ be an integer. We define the Turaev–Viro invariant $\mathrm {TV}_{r}(\mathfrak {T})$ as follows.

Let V, E, F and T denote the set of vertices, edges, triangles and tetrahedra respectively of the triangulation $\mathfrak {T}$. Let $I = \{0, 1/2, 1, 3/2, \ldots , (r-2)/2\}$; note that $|I|=r-1$. We define a colouring of $\mathfrak {T}$ to be a map $\theta : E \rightarrow I$; that is, $\theta$ “colours” each edge of $\mathfrak {T}$ with an element of I. A colouring $\theta$ is admissible if, for each triangle of $\mathfrak {T}$, the three edges $e_1$, $e_2$, and $e_3$ bounding the triangle satisfy:

• the parity condition $\theta (e_1)+\theta (e_2)+\theta (e_3)\in \mathbb {Z}$;

• the triangle inequalities $\theta (e_1) \le \theta (e_2) + \theta (e_3)$, $\theta (e_2) \le \theta (e_1) + \theta (e_3)$, and $\theta (e_3) \le \theta (e_1) + \theta (e_2)$; and

• the upper bound constraint $\theta (e_1)+\theta (e_2)+\theta (e_3)\le r-2$.

More generally, we refer to any triple $(i,j,k) \in I \times I \times I$ satisfying these three conditions as an admissible triple of colours.

For each admissible colouring $\theta$ and for each vertex $v \in V$, edge $e \in E$, triangle $f \in F$ or tetrahedron $t \in T$, we define weights $|v|_\theta , |e|_\theta , |f|_\theta , |t|_\theta \in \mathbb {C}$. The precise definition of these weights is rather involved and explained in detail below.

Our notation differs slightly from Turaev and Viro (1992); most notably, Turaev and Viro do not consider triangle weights $|f|_\theta$, but instead incorporate an additional factor of $|f|_\theta ^{1/2}$ into each tetrahedron weight $|t|_\theta$ and $|t'|_\theta$ for the two tetrahedra t and $t'$ containing f. This choice of notation simplifies the notation and avoids unnecessary (but harmless) ambiguities when taking square roots.

Let $\zeta = e^{i \pi s / r} \in \mathbb {C}$, $0<s<2r$, $\gcd (r,s) = 1$. Note that this implies that $\zeta$ is a (2r)th root of unity, and that $\zeta ^2$ is a primitive rth root of unity; that is, $(\zeta ^2)^k \ne 1$ for $k=1,\ldots ,r-1$. For each positive integer i, we define $[i] = (\zeta ^i-\zeta ^{-i})/(\zeta -\zeta ^{-1})$ and, as a special case, $[0]= 1$. We next define the “bracket factorial” $[i]! = [i]\,[i-1] \ldots [0]$. Note that $[r] = 0$, and thus $[i]! = 0$ for all $i \ge r$.

We give every vertex constant weight

$$\begin{aligned} |v|_\theta = \frac{\left| \zeta -\zeta ^{-1}\right| ^2}{2r}, \end{aligned}$$

and to each edge e of colour $i \in I$ (i.e., for which $\theta (e) = i$) we give the weight

$$\begin{aligned} |e|_\theta = (-1)^{2i} \cdot [2i+1]. \end{aligned}$$

A triangle f whose three edges have colours $i,j,k \in I$ is assigned the weight

$$\begin{aligned} |f|_\theta = (-1)^{i+j+k} \cdot \frac{[i+j-k]! \cdot [i+k-j]! \cdot [j+k-i]!}{[i+j+k+1]!}. \end{aligned}$$

Note that the parity condition and triangle inequalities ensure that the argument inside each bracket factorial is a non-negative integer.

Fig. 1

Edge colours of a tetrahedron

Finally, let t be a tetrahedron with edges colours $i_0,i_1,i_2,i_3,i_4,i_5$ as indicated in Fig. 1. In particular, the four triangles surrounding t have colours $(i_0,i_1,i_3)$, $(i_0,i_2,i_4)$, $(i_1,i_2,i_5)$ and $(i_3,i_4,i_5)$, and the three pairs of opposite edges have colours $(i_0,i_5)$, $(i_1,i_4)$ and $(i_2,i_3)$. We define

$$\begin{aligned} \tau _\phi (t,z)= & {} [z-i_0-i_1-i_3]! \cdot [z-i_0-i_2-i_4]! \cdot [z-i_1-i_2-i_5]! \\&\cdot \, [z-i_3-i_4-i_5]!, \\ \kappa _\phi (t,z)= & {} [i_0+i_1+i_4+i_5-z]! \cdot [i_0+i_2+i_3+i_5-z]! \cdot [i_1+i_2+i_3+i_4-z]! \end{aligned}$$

for all integers z such that the bracket factorials above all have non-negative arguments; equivalently, for all integers z in the range $z^-\le z \le z^+$ with

$$\begin{aligned} z^-= & {} \max \{i_0+i_1+i_3,\ i_0+i_2+i_4,\ i_1+i_2+i_5,\ i_3+i_4+i_5\}; \\ z^+= & {} \min \{i_0+i_1+i_4+i_5,\ i_0+i_2+i_3+i_5,\ i_1+i_2+i_3+i_4\}. \end{aligned}$$

Note that, as before, the parity condition ensures that the argument inside each bracket factorial above is an integer. We then declare the weight of tetrahedron t to be

$$\begin{aligned} |t|_\phi = \sum _{z^-\le z \le z^+} \frac{(-1)^z \cdot [z+1]!}{\tau _\phi (t,z) \cdot \kappa _\phi (t,z)}, \end{aligned}$$

Note that all weights are polynomials in $\zeta$ with rational coefficients.

Using these weights, we define the weight of the colouring to be

$$\begin{aligned} |\mathfrak {T}|_\theta = \prod _{v \in V} |v|_\theta \times \prod _{e \in E} |e|_\theta \times \prod _{f \in F} |f|_\theta \times \prod _{t \in T} |t|_\theta , \end{aligned}$$

(1)

and the Turaev–Viro invariant to be the sum over all admissible colourings

$$\begin{aligned} \mathrm {TV}_{r}(\mathfrak {T}) = \sum _{\theta \ \mathrm {admissible}} |\mathfrak {T}|_\theta . \end{aligned}$$

(2)

Turaev and Viro (1992) show that $\mathrm {TV}_{r}(\mathfrak {T})$ is indeed an invariant of the manifold; that is, if $\mathfrak {T}$ and $\mathfrak {T}'$ are generalised triangulations of the same closed 3-manifold M, then $\mathrm {TV}_{r}(\mathfrak {T}) = \mathrm {TV}_{r}(\mathfrak {T}')$ for all r. Although $\mathrm {TV}_{r}(\mathfrak {T})$ is defined on the complex numbers $\mathbb {C}$, it always takes a real value (it is equal to the square of the modulus of a Witten–Reshetikhin–Turaev invariant) (Walker 1991).

In Sect. 5 we describe the philosophy of the Turaev–Viro–Barrett–Westbury construction of topological invariants (Barrett and Westbury 1996), which substantially generalises the Turaev–Viro model.

2.2 Treewidth and parameterized complexity

Throughout this paper we always refer to nodes and arcs of graphs, to clearly distinguish these from the vertices and edges of triangulations.

Robertson and Seymour (1986) introduced the concept of the treewidth of a graph, which now plays a major role in parameterized complexity. Here, we adapt this concept to triangulations in a straightforward way.

Definition 1

Let $\mathfrak {T}$ be a generalised triangulation of a 3-manifold, and let T be the set of tetrahedra in $\mathfrak {T}$. A tree decomposition $(X, \{B_\tau \})$ of $\mathfrak {T}$ consists of a tree X and bags $B_\tau \subseteq T$ for each node $\tau$ of X, for which:

• each tetrahedron $t \in T$ belongs to some bag $B_\tau$;

• if a face of some tetrahedron $t_1 \in T$ is identified with a face of some other tetrahedron $t_2 \in T$, then there exists a bag $B_\tau$ with $t_1,t_2 \in B_\tau$;

• for each tetrahedron $t \in T$, the bags containing t correspond to a connected subtree of X.

The width of this tree decomposition is defined as $\max |B_\tau |-1$. The treewidth of $\mathfrak {T}$, denoted $\mathrm {tw}(\mathfrak {T})$, is the smallest width of any tree decomposition of $\mathfrak {T}$.

The relationship between this definition and the classical graph-theoretical notion of treewidth is simple: $\mathrm {tw}(\mathfrak {T})$ is the treewidth of the dual graph of $\mathfrak {T}$, the 4-valent multigraph whose nodes correspond to tetrahedra of $\mathfrak {T}$ and whose arcs represent pairs of tetrahedron faces that are identified together.

Figure 2 shows the dual graph of a 9-tetrahedra triangulation of a 3-manifold, along with a possible tree decomposition. The largest bags have size three, and so the width of this tree decomposition is $3-1=2$.

Fig. 2

The dual graph and a tree decomposition of a 3-manifold triangulation

Definition 2

A nice tree decomposition of a generalised triangulation $\mathfrak {T}$ is a tree decomposition $(X, \{B_\tau \})$ of $\mathfrak {T}$ whose underlying tree X is rooted, and where:

• The bag $B_\rho$ at the root of the tree is empty ($B_{\rho }$ is called the root bag);

• If a bag $B_{\tau }$ has no children, then $|B_{\tau }|=1$ (such a $B_{\tau }$ is called a leaf bag);

• If a bag $B_{\tau }$ has two children $B_{\sigma }$ and $B_{\mu }$, then $B_{\tau } = B_{\sigma } = B_{\mu }$ (such a $B_{\tau }$ is called a join bag);

• Every other bag $B_{\tau }$ has precisely one child $B_{\sigma }$, and either:

  • $|B_{\tau }| = |B_{\sigma }| + 1$ and $B_{\tau } \supset B_{\sigma }$ (such a $B_{\tau }$ is called an introduce bag), or

  • $|B_{\tau }| = |B_{\sigma }| - 1$ and $B_{\tau } \subset B_{\sigma }$ (such a $B_{\tau }$ is called a forget bag).

Given a tree decomposition of a triangulation $\mathfrak {T}$ of width k and O(n) bags, we can convert this in O(n) time into a nice tree decomposition of $\mathfrak {T}$ that also has width k and O(n) bags (Kloks 1994).

3 Algorithms for computing Turaev–Viro invariants

All of the algorithms in this paper use exact arithmetic. This is crucial if we wish to avoid floating-point numerical instability, since computing $\mathrm {TV}_{r}$ may involve exponentially many arithmetic operations.

We briefly describe how this exact arithmetic works. Since all weights in the definition of $\mathrm {TV}_{r}$ are rational polynomials in $\zeta = e^{i \pi s/r}$, $0<s<2r$, all arithmetic operations remain within the rational field extension $\mathbb {Q}(\zeta )$. If $\zeta$ is a primitive nth root of unity then this field extension is called the nth cyclotomic field. This in turn is isomorphic to the polynomial field $\mathbb {Q}[X] / \varPhi _n(X)$, where $\varPhi _n(X)$ is the nth cyclotomic polynomial with degree $\varphi (n)$ (Euler’s totient function). Therefore we can implement exact arithmetic using degree $\varphi (n)$ polynomials over $\mathbb {Q}$.

If r is odd and s is even, then $\zeta$ is a primitive rth root of unity, and $\mathbb {Q}(\zeta ) \cong \mathbb {Q}[X] / \varPhi _r(X)$. Otherwise $\zeta$ is a primitive (2r)th root of unity, and $\mathbb {Q}(\zeta ) \cong \mathbb {Q}[X] / \varPhi _{2r}(X)$. In this paper we give our complexity results in terms of arithmetic operations in $\mathbb {Q}(\zeta )$.

Let $\zeta$ be an n-root of unity and $\mathbb {Q}(\zeta )$ be the nth cyclotomic field. We represent elements of $\mathbb {Q}(\zeta )$ by polynomials of degree at most $\varphi (n)$, with rational coefficients, using the isomorphism $\mathbb {Q}(\zeta ) \cong \mathbb {Q}[X] / \varPhi _n(X)$. Asymptotically, the Euler totient function satisfies $\varphi (n) = \varTheta (n)$. Additions of two polynomials of degree at most n are performed in O(n) operations in $\mathbb {Q}$, and multiplications and divisions are performed in O(M(n)) operations in $\mathbb {Q}$, with $M(n) = O(n \log n \log \log n)$ (Cantor et al. 1991).

Hence, for fixed r, Turaev–Viro invariants can be computed in $O(N(r) \cdot r \log r \log \log r)$ operations in $\mathbb {Q}$ using exact arithmetic over cyclotomic fields, where N(r) denotes the number of arithmetic operations needed to compute $\mathrm {TV}_{r}$.

3.1 The backtracking algorithm for computing $\mathrm {TV}_{r}$

There is a straightforward but slow algorithm to compute $\mathrm {TV}_{r}$ for arbitrary r. The core idea is to use a backtracking algorithm to enumerate all admissible colourings of edges, and compute and sum their weights. Both major software packages that compute Turaev–Viro invariants—the Manifold Recogniser (Matveev et al. 2012) and Regina (Burton et al. 1999)—currently employ optimised variants of this.

Let $\mathfrak {T}$ be a 3-manifold triangulation, with $\ell$ edges $e_1, \ldots , e_\ell$. A simple Euler characteristic argument gives $\ell = n + v$ where n is the number of tetrahedra and v is the number of vertices in $\mathfrak {T}$. Therefore $\ell \in \varTheta (n)$.

To enumerate colourings, since each edge admits $r-1$ possible colours, the backtracking algorithm traverses a search tree of $O((r-1)^\ell )$ nodes: a node at depth i corresponds to a partial colouring of the edges $e_1, \ldots , e_i$, and each node has degree $r-1$ (one edge per colour). Each leaf on the tree represents a (possibly not admissible) colouring of all the edges. At each node we maintain the weight of the current partial colouring, and updates this weight as we traverse the tree. If we reach a leaf whose colouring is admissible, we add this weight to our total.

Lemma 1

If we sort the edges $e_1, \ldots ,e_\ell$ by decreasing degree, the backtracking algorithm terminates in $O((r-1)^\ell )$ arithmetic operations in $\mathbb {Q}(\zeta )$.

Proof

The proof is simple. The main complication is to ensure that updating the weight of the current partial colouring takes amortised constant time. For this we use Chebyshev’s inequality, plus the observation that the average edge degree is $\le 6$.

In more detail, suppose that the edges $e_1, \ldots ,e_\ell$ are ordered by decreasing degree. Let $\deg (e_i)$ be the degree of edge $e_i$. Changing the colour of $e_i$ affects the colours of the $\deg (e_i)$ triangles and $\deg (e_i)$ tetrahedra containing $e_i$. Hence the update of the current partial colouring weight is performed in $O(\deg (e_i))$ arithmetic operations in $\mathbb {Q}(\zeta )$. The total number of arithmetic operations performed by the algorithm is consequently $O(\sum _i (r-1)^i \deg (e_i))$. Following an Euler characteristic argument, a triangulation of a closed 3-manifold with $\ell$ edges and v vertices has $n=\ell -v$ tetrahedra and, consequently, the average degree of an edge is $6(\ell -v)/\ell$ and thus constant. Considering that the sequence $((r-1)^i)_i$ is increasing and $\deg e_i$ is decreasing, we conclude using Chebyshev’s sum inequality that

$$\begin{aligned} O\left( \sum _i (r-1)^i \deg (e_i)\right) = O\left( \sum _i (r-1)^i\right) = O((r-1)^\ell ). \end{aligned}$$

To obtain a bound in the number of tetrahedra n, we note that a closed and connected 3-manifold triangulation with $n > 2$ tetrahedra must have $v \le n+1$ vertices. Combined with $n=\ell -v$ above, we have a worst-case running time of $O((r-1)^{2n+1})$ arithmetic operations in $\mathbb {Q}(\zeta )$.

3.2 A polynomial-time algorithm for $r=3$

Throughout this section, $\mathfrak {T}$ denotes an n-tetrahedra triangulation of an orientable 3-manifold M.

Remark 1

We introduce homology with coefficients in the field $\mathbb {Z}_2$. A generalised triangulation $\mathfrak {T}$, after gluing, contains a set of vertices (dimension 0), edges (dimension 1), triangles (dimension 2) and tetrahedra (dimension 3) with incidence relations. The group of d-chains of $\mathfrak {T}$, $d \in \{0, \ldots ,3\}$, denoted by $\mathbf {C}_d(\mathfrak {T},\mathbb {Z}_2)$ is the group of formal sums of d-simplices with $\mathbb {Z}_2$ coefficients. The dth boundary operator is a linear operator $\partial _d: \mathbf {C}_d(\mathfrak {T},\mathbb {Z}_2) \rightarrow \mathbf {C}_{d-1}(\mathfrak {T},\mathbb {Z}_2)$ that assigns to a d-face the alternate sum of its boundary faces. The kernel of $\partial _d$, denoted by $\mathbf {Z}_d(\mathfrak {T},\mathbb {Z}_2)$, is the group of d-cycles and the image of $\partial _d$, denoted by $\mathbf {B}_{d-1}(\mathfrak {T},\mathbb {Z}_2)$, is the group of $(d-1)$-boundaries. The fundamental property of homology is that $\partial _d \circ \partial _{d+1}=0$, and we define the dth homology group $\mathbf {H}_d(\mathfrak {T},\mathbb {Z}_2)$ of $\mathfrak {T}$ to be the quotient $\mathbf {H}_d(\mathfrak {T},\mathbb {Z}_2) = \mathbf {Z}_d(\mathfrak {T},\mathbb {Z}_2)/\mathbf {B}_d(\mathfrak {T},\mathbb {Z}_2)$. It is a finite dimensional $\mathbb {Z}_2$-vector space and we denote its dimension by $\beta _d(\mathfrak {T},\mathbb {Z}_2)$.

For a more thorough introduction into homology theory, see Munkres (1984).

There are two Turaev–Viro invariants for $r=3$, at $\zeta = e^{i \pi / 3}$ and at $\zeta = e^{i \pi 2 / 3}$. In both cases the value of $\mathrm {TV}_{3}(\mathfrak {T})$ is closely related to $\mathbf {H}_2(M,\mathbb {Z}_2)$, the 2- dimensional homology group of M with $\mathbb {Z}_2$ coefficients. $\mathbf {H}_2(M, \mathbb {Z}_2)$ is a $\mathbb {Z}_2$-vector space whose dimension is the second Betti number $\beta _2 (M, \mathbb {Z}_2)$. Its elements are (for our purposes) equivalence classes of 2-cycles, called homology classes, which can be represented by 2-dimensional triangulated surfaces S embedded in $\mathfrak {T}$.

The Euler characteristic of a triangulated surface S, denoted by $\chi (S)$, is $\chi (S) = v-e+f$, where v, e and f denote the number of vertices, edges and triangles of S respectively. We define the Euler characteristic $\chi (c)$ of a 2-cycle c to be the Euler characteristic of the embedded surface it represents. Given $\mathfrak {T}$, the dimension $\beta _2 (M, \mathbb {Z}_2)$ of $\mathbf {H}_2(M,\mathbb {Z}_2)$ may be computed in $O(\mathrm {poly}(n))$ operations.

The following result is well known (Matveev 2003):

Proposition 1

Let M be a closed orientable 3-manifold. If M contains a 2-cycle with odd Euler characteristic, then $\mathrm {TV}_{3}(M) = \text {the order of } \mathbf {H}_2(M,\mathbb {Z}_2)$ at $\zeta = e^{i \pi 2 / 3}$ and $\mathrm {TV}_{3}(M) = 0$ at $\zeta = e^{i \pi / 3}$. Otherwise we have $\mathrm {TV}_{3}(M) = \text {the order of } \mathbf {H}_2(M,\mathbb {Z}_2)$ for both $\zeta = e^{i \pi / 3}$ and $\zeta = e^{i \pi 2 / 3}$.

Consequently $\mathrm {TV}_{3}(M) = 2^{\beta _2(M,\mathbb {Z}_2)}$ at $\zeta = e^{i \pi 2 / 3}$, and one can compute $\mathrm {TV}_{3}(M)$ in this case in polynomial time. The parity of the Euler characteristic of 2-cycles does not change within a homology class; moreover, given two 2-cycles c and $c'$, $\chi (c + c') \equiv \chi (c) + \chi (c') \ \mathrm {mod} \ 2$. Consequently, for $\zeta = e^{i \pi / 3}$, one can check whether $\mathrm {TV}_{3}(M)=0$ or $\mathrm {TV}_{3}(M) = \text {the order of } \mathbf {H}_2(M,\mathbb {Z}_2)$ by computing the Euler characteristic of a cycle in each of the $\beta _2(M,\mathbb {Z}_2)$ homology classes that generate $\mathbf {H}_2(M,\mathbb {Z}_2)$. Because $\beta _2(M,\mathbb {Z}_2) = O(n)$, this leads to a polynomial time algorithm also.

3.3 $\#$P-hardness of $\mathrm {TV}_{4}$

The complexity class #P is a function class that counts accepting paths of a non-deterministic Turing machine (Valiant 1979). Informally, given an NP decision problem C asking for the existence of a solution, its #P analogue $\#C$ is a counting problem asking for the number of such solutions. A problem is #P-hard if every problem in $\#$P polynomially reduces to it. For example, the problem $\#3SAT$, which asks for the number of satisfying assignments of a 3CNF formula, is $\#$P-hard.

Naturally, counting problems are “harder” than their decision counterpart, and so $\#$P-hard problems are at least as hard as NP-complete problems—specifically, $\#$P-complete problems are as hard as any problem in the polynomial hierarchy (Toda 1991). Hence proving $\#$P-hardness is a strong complexity statement.

Kirby and Melvin (2004) prove that computing the Witten–Reshetikhin–Turaev invariant $\tau _r$ is $\#$P-hard for $r=4$. This invariant $\tau _r$ is a more complex 3-manifold invariant which is closely linked to the Turaev–Viro invariant $\mathrm {TV}_{r}$ at $\zeta = e^{i \pi / r}$ by the formula $\mathrm {TV}_{r}(M) = |\tau _r(M)|^2$. Although computing $\mathrm {TV}_{r}$ is “easier” than computing $\tau _r$, we can use the Kirby-Melvin hardness proof to fit our purposes.

To prove their result, Kirby and Melvin reduce the problem of counting the zeros of a cubic form to the computation of $\tau _4$. Given a cubic form

$$\begin{aligned} c(x_1, \ldots ,x_n) = \sum _i c_i\ x_i + \sum _{i,j} c_{ij}\ x_i x_j + \sum _{i,j,k} c_{ijk}\ x_i x_j x_k \end{aligned}$$

in n variables over $\mathbb {Z}_2$ and with $\#c$ zeros, they define a triangulation of a 3-manifold $M_c$ with $O(\mathrm {poly}(n))$ tetrahedra satisfying $\tau _4(M_c) = 2 \#c - 2^n$ and hence $\mathrm {TV}_{r} = (2 \#c - 2^n)^2$.

Consequently, counting the zeros of $c(x_1, \ldots ,x_n)$ reduces to computing $\tau _4(M_c)$, and so computing $\mathrm {TV}_{4}$ determines $\#c$ up to a ± sign ambiguity (depending on whether or not c admits more than half of the input as zeros).

Establishing the existence of a zero for a cubic form is an NP-complete problem, which implies that counting the number of zeros is $\#$P-complete. Consequently, computing $\tau _4$ is #P-hard. Kirby and Melvin prove this claim by reducing $\#3SAT$ to the problem of counting the zeros of a cubic form; moreover, we observe that their construction ensures that this cubic form admits more than half of its inputs as zeros.

We recall the reduction of $\#3SAT$ to the problem of counting the number of zeros of a cubic form over $\mathbb {Z}_2$ found in Kirby and Melvin (2004). Given a 3CNF formula over variables $x_1, \ldots , x_n$:

$$\begin{aligned} C = C_1 \wedge \cdots \wedge C_m\quad \text {with} \ C_j = \overline{x}_{j_1} \vee \overline{x}_{j_2} \vee \overline{x}_{j_3} \end{aligned}$$

and $\overline{x}_i$ is either $x_i$ or its negation $\lnot x_i$ the problem $\#3SAT$ consists in counting the number of assignments of “true” and “false” to the variables $x_1, \ldots ,x_n$ satisfying the formula.

To each form $C_j = \overline{x}_{j_1} \vee \overline{x}_{j_2} \vee \overline{x}_{j_3}$ we assign a cubic equation $q_j$ over $\mathbb {Z}_2$ by setting $\text {``true''} = 0$ and $\text {``false''} = 1$, and replacing $x_i$ by the variable $x_i$ and $\lnot x_i$ by $(1-x_i)$. For example, a form $(\lnot x_i \wedge \lnot x_j \wedge x_k)$ leads to the equation $(1-x_i)(1-x_j)x_k = 0$. An assignment satisfies $C_j$ if and only if it cancels $q_j$, hence the number of solution to the system of equations $\{ q_1(x_1, \ldots ,x_n) = 0, \ldots ,q_m(x_1,\ldots ,x_n) = 0\}$ is equal to $\#c$, the number of satisfying assignments for C.

We turn each cubic equation into two quadratic equations by introducing a new variable $x_{ij}$ for each monomial $x_i x_j x_k$ of degree 3 and a new quadratic equation $x_{ij} - x_i x_j=0$, and by replacing the product $x_i x_j x_k$ by $x_{ij} x_k$. We obtain a set of $m'$ equations $\{\overline{q_1}(x_1, \ldots ,x_n) = 0, \ldots , \overline{q_{m'}}(x_1,\ldots ,x_{n'}) = 0\}$ in $n'$ variables over $\mathbb {Z}_2$, with $m < m' \le 2m$ and $n < n' \le 2n$. The number of solutions of this system remains $\#c$.

Finally, define the following cubic form c by introducing $m'$ extra variables $z_1, \ldots , z_{m'}$:

$$\begin{aligned} Q = \sum _{i=1}^{m'}z_i \overline{q_i}(x_1, \ldots ,x_{n'}). \end{aligned}$$

The number of zeros of Q is equal to $2^{m'} \#c + 2^{m'-1}(2^{n'}-\#c) \ge \frac{1}{2} 2^{n'+m'}$. Because Q is defined on $n'+m'$ variables it admits more than half of its input as zeros. Finally, $\#3SAT$ reduces to counting the number of zeros of a cubic form which admits at least half of its input as zeros.

Thus the same reduction process as for $\tau _4$ applies for $\mathrm {TV}_{4}$, and so:

Corollary 1

Computing $\mathrm {TV}_{4}$ at $\zeta = e^{i \pi / 4}$ is $\#$P-hard.

Remark 2

Very recently, a subset of the authors refined this complexity study of $\mathrm {TV}_{4}$ in Maria and Spreer (2017). More precisely, given an n-tetrahedra triangulation $\mathfrak {T}$ of a 3-manifold M, they present an algorithm to compute $\mathrm {TV}_{4} (M)$ in running time $O(2^{\beta _1(M,\mathbb {Z}_2)}n^3)$. Note that, in particular, the exponential part of the running time is independent of the triangulation $\mathfrak {T}$.

However, the methods to obtain this result fail to be applicable to $\mathrm {TV}_{r}$, $r > 4$.

4 A fixed-parameter tractable algorithm for Turaev–Viro invariants

Here, we present an explicit algorithm for computing the Turaev–Viro invariants $\mathrm {TV}_{r}$ of a 3-manifold triangulation, r fixed, which is FPT in the treewidth of the dual graph of the input triangulation. As is common for treewidth-based methods, the algorithm involves dynamic programming over a tree decomposition $(X,\{B_\tau \})$. We first describe the data that we compute and store at each bag $B_\tau$, and then give the algorithm itself.

Our first step is to reorganise the formula for $\mathrm {TV}_{r}(\mathfrak {T})$ to be a product over tetrahedra only. This makes it easier to work with “partial colourings” corresponding to subsets of edges $\mathfrak {T}$.

Definition 3

Let $\mathfrak {T}$ be a generalised triangulation of a 3-manifold, and let V, E, F and T denote the vertices, edges, triangles and tetrahedra of $\mathfrak {T}$ respectively. For each vertex $x \in V$, each edge $x \in E$ and each triangle $x \in F$, we arbitrarily choose some tetrahedron $\varDelta (x)$ that contains x.

Now consider the definition of $\mathrm {TV}_{r}(\mathfrak {T})$. For each admissible colouring $\theta : E \rightarrow I$ and each tetrahedron $t \in T$, we define the adjusted tetrahedron weight $|t|'_{\theta }$:

$$\begin{aligned} |t|'_\theta = |t|_\theta \times \prod _{{\mathop {\varDelta (v)=t}\limits ^{v \in V}}} |v|_\theta \times \prod _{{\mathop {\varDelta (e)=t}\limits ^{e \in E}}} |e|_\theta \times \prod _{{\mathop {\varDelta (f)=t}\limits ^{f \in F}}} |f|_\theta . \end{aligned}$$

It follows from Eq. (1) that the full weight of the colouring $\theta$ is just

$$\begin{aligned} |\mathfrak {T}|_\theta = \prod _{t \in T} |t|'_\theta . \end{aligned}$$

Notation 1

Let X be a rooted tree. For any non-root node $\tau$ of X, we denote the parent node of $\tau$ by $\hat{\tau }$. For any two nodes $\sigma ,\tau$ of X, we write $\sigma \prec \tau$ if $\sigma$ is a descendant node of $\tau$.

Definition 4

Let $\mathfrak {T}$ be a generalised triangulation of a 3-manifold, and let V, E, F and T denote the vertices, edges, triangles and tetrahedra of $\mathfrak {T}$ respectively. Let $(X, \{B_\tau \})$ be a nice tree decomposition of $\mathfrak {T}$. For each node $\tau$ of the rooted tree X, we define the following sets:

• $T_\tau \subseteq T$ is the set of all tetrahedra that appear in bags beneath $\tau$ but not in the bag $B_\tau$ itself. More formally: $T_\tau = (\bigcup _{\sigma \prec \tau } B_\sigma ) \backslash B_\tau$.

• $F_\tau \subseteq F$ is the set of all triangles that appear in some tetrahedron $t \in T_\tau$.

• $E_\tau \subseteq E$ is the set of all edges that appear in some tetrahedron $t \in T_\tau$.

• $E^*_\tau \subseteq E_\tau$ is the set of all edges that appear in some tetrahedron $t \in T_\tau$ and also some other tetrahedron $t' \notin T_\tau$; we refer to these as the current edges at node $\tau$.

We can make the following immediate observations:

Lemma 2

If $\tau$ is a leaf of the tree X, then we have $T_\tau = F_\tau = E_\tau = E^*_\tau = \emptyset$. If $\tau$ is the root of the tree X, then we have $T_\tau = T$, $F_\tau = F$, $E_\tau = E$, and $E^*_\tau = \emptyset$.

The key idea is, at each node $\tau$ of the tree, to store explicit colours on the “current” edges $e \in E^*_\tau$ and to aggregate over all colours on the “finished” edges $e \in E_\tau \backslash E^*_\tau$. For this we need some further definitions and notation.

Definition 5

Again let $\mathfrak {T}$ be a generalised triangulation of a 3-manifold, and let $(X, \{B_\tau \})$ be a nice tree decomposition of $\mathfrak {T}$. Fix some integer $r \ge 3$, and consider the set of colours $I = \{0, 1/2, 1, 3/2, \ldots , (r-2)/2\}$ as used in defining the Turaev–Viro invariants $\mathrm {TV}_{r}$.

Let $\tau$ be any node of X. We examine “partial colourings” that only assign colours to the edges in $E_\tau$ or $E^*_\tau$:

• Consider any colouring $\theta : E_\tau \rightarrow I$. We call $\theta$ admissible if, for each triangle in $F_\tau$, the three edges e, f, g bounding the triangle yield an admissible triple $(\theta (e), \theta (f), \theta (g))$.

• Define $\varPsi _\tau$ to be the set of all colourings $\psi : E^*_\tau \rightarrow I$ that can be extended to any admissible colouring $\theta : E_\tau \rightarrow I$.

• Consider any colouring $\psi \in \varPsi _\tau$ (so $\psi : E^*_\tau \rightarrow I$). We define the “partial invariant”

  $$\begin{aligned} \mathrm {TV}_{r}(\mathfrak {T},\tau ,\psi ) = \sum _{{\mathop {\theta = \psi \ \mathrm{{on}}\ E^*_\tau }\limits ^{\theta \ \mathrm{{admissible}}}}} \prod _{t \in T_\tau } |t|'_\theta . \end{aligned}$$

Essentially, the partial invariant $\mathrm {TV}_{r}(\mathfrak {T},\tau ,\psi )$ considers all admissible ways $\theta$ of extending the colouring $\psi$ from the current edges $E^*_\tau$ to also include the “finished” edges in $E_\tau$, and then sums the partial weights $|t|'_\theta$ for all such extensions $\theta$ using only the tetrahedra in $T_\tau$.

We can now give a description of our algorithm for $\mathrm {TV}_{r}$.

Algorithm 1

Let $\mathfrak {T}$ be a generalised triangulation of a 3-manifold. We compute $\mathrm {TV}_{r}(\mathfrak {T})$ for given r and a given 2r-th root of unity $\zeta$ as follows.

Build a nice tree decomposition $(X, \{B_\tau \})$ of $\mathfrak {T}$. Then work through each node $\tau$ of X from the leaves of X to the root, and compute $\varPsi _\tau$ and $\mathrm {TV}_{r}(\mathfrak {T},\tau ,\psi )$ for each $\psi \in \varPsi _\tau$ as follows.

• 1. If $\tau$ is a leaf bag, then $E^*_\tau = E_\tau = \emptyset$, $\varPsi _\tau$ contains just the trivial colouring $\psi$ on $\emptyset$, and $\mathrm {TV}_{r}(\mathfrak {T},\tau ,\psi ) = 1$.

• 2. If $\tau$ is some other introduce bag with child node $\sigma$, then $T_\tau = T_\sigma$. This means that $\varPsi _\tau = \varPsi _\sigma$, and for each $\psi \in \varPsi _\tau$ we have $\mathrm {TV}_{r}(\mathfrak {T},\tau ,\psi ) = \mathrm {TV}_{r}(\mathfrak {T},\sigma ,\psi )$.

• 3. If $\tau$ is a forget bag with child node $\sigma$, then $T_\tau = T_\sigma \cup \{t\}$ for the unique “forgotten” tetrahedron $t \in B_\tau \backslash B_\sigma$. Moreover, $E^*_\tau$ extends $E^*_\sigma$ by including the six edges of t (if they were not already present). For each colouring $\psi \in \varPsi _\sigma$, enumerate all possible ways of colouring the six edges of t that are consistent with $\psi$ on any edges of t that already appear in $E^*_\sigma$, and are admissible on the four triangular faces of t. Each such colouring on t yields an extension $\psi ' : E^*_\tau \rightarrow I$ of $\psi : E^*_\sigma \rightarrow I$. We include $\psi '$ in $\varPsi _\tau$, and record the partial invariant $\mathrm {TV}_{r}(\mathfrak {T},\tau ,\psi ') = \mathrm {TV}_{r}(\mathfrak {T},\sigma ,\psi )$.

• 4. If $\tau$ is a join bag with child nodes $\sigma _1,\sigma _2$, then $T_\tau$ is the disjoint union $T_{\sigma _1} \mathbin {\dot{\cup }}T_{\sigma _2}$. Here $E^*_\tau$ is a subset of $E^*_{\sigma _1} \cup E^*_{\sigma _2}$. For each pair of colourings $\psi _1 \in \varPsi _{\sigma _1}$ and $\psi _2 \in \varPsi _{\sigma _2}$, if $\psi _1$ and $\psi _2$ agree on the common edges in $E^*_{\sigma _1} \cap E^*_{\sigma _2}$ then record the pair $(\psi _1,\psi _2)$. Each such pair yields a “combined colouring” in $\varPsi _\tau$, which we denote by $\psi _1 \cdot \psi _2 : E^*_\tau \rightarrow I$; note that different pairs $(\psi _1,\psi _2)$ might yield the same colouring $\psi _1 \cdot \psi _2$ since some edges from $E^*_{\sigma _1} \cup E^*_{\sigma _2}$ might not appear in $E^*_\tau$. Then $\varPsi _\tau$ consists of all such combined colourings $\psi _1 \cdot \psi _2$ from recorded pairs $(\psi _1,\psi _2)$. Moreover, for each combined colouring $\psi \in \varPsi _\tau$ we compute the partial invariant $\mathrm {TV}_{r}(\mathfrak {T},\tau ,\psi )$ by aggregating over all duplicates:

  $$\begin{aligned} \mathrm {TV}_{r}(\mathfrak {T},\tau ,\psi ) = \sum _{{\mathop {\psi _1 \cdot \psi _2 = \psi }\limits ^{(\psi _1,\psi _2)\ {\mathrm {recorded}}}}} \mathrm {TV}_{r}(\mathfrak {T},\sigma _1,\psi _1) \cdot \mathrm {TV}_{r}(\mathfrak {T},\sigma _2,\psi _2). \end{aligned}$$

Once we have processed the entire tree, the root node $\rho$ of X has $E^*_\rho = \emptyset$, $\varPsi _\rho$ contains just the trivial colouring $\psi$ on $\emptyset$, and $\mathrm {TV}_{r}(\mathfrak {T},\rho ,\psi )$ for this trivial colouring is equal to the Turaev–Viro invariant $\mathrm {TV}_{r}(\mathfrak {T})$.

The time complexity of this algorithm is simple to analyse. Each leaf bag or introduce bag can be processed in O(1) time (of course for the introduce bag we must avoid a deep copy of the data at the child node). Each forget bag produces $|\varPsi _\tau | \le (r-1)^{|E^*_\tau |}$ colourings, each of which takes $O(|E^*_\tau |)$ time to analyse.

Naïvely, each join bag requires us to process $|\varPsi _{\sigma _1}| \cdot |\varPsi _{\sigma _2}| \le (r-1)^{|E^*_{\sigma _1}| + |E^*_{\sigma _2}|}$ pairs of colourings $(\psi _1,\psi _2)$. However, we can optimise this. Since we are only interested in colourings that agree on $E^*_{\sigma _1} \cap E^*_{\sigma _2}$, we can first partition $\varPsi _{\sigma _1}$ and $\varPsi _{\sigma _2}$ into buckets according to the colours on $E^*_{\sigma _1} \cap E^*_{\sigma _2}$, and then combine pairs from each bucket individually. This reduces our work to processing at most $(r-1)^{|E^*_{\sigma _1} \cup E^*_{\sigma _2}|}$ pairs overall. Each pair takes $O(|E^*_\tau |)$ time to process, and the preprocessing cost for partitioning $\varPsi _{\sigma _i}$ is $O(|\varPsi _{\sigma _i}| \cdot \log |\varPsi _{\sigma _i}| \cdot |E^*_{\sigma _i}|) = O((r-1)^{|E^*_{\sigma _i}|} \cdot |E^*_{\sigma _i}|^2 \log r)$.

Suppose that our tree decomposition has width k. At each tree node $\tau$, every edge in $E^*_\tau$ must belong to some tetrahedron in the bag $B_\tau$, and so $|E^*_\tau | \le 6(k+1)$. Likewise, at each join bag described above, every edge in $E^*_{\sigma _1}$ or $E^*_{\sigma _2}$ must belong to some tetrahedron in the bag $B_{\sigma _i}$ and therefore also the parent bag $B_\tau$, and so $|E^*_{\sigma _1} \cup E^*_{\sigma _2}| \le 6(k+1)$. From the discussion above, it follows that every bag can be processed in time $O\left( (r-1)^{6(k+1)} \cdot k^2 \log r\right)$, and so:

Theorem 1

Given a generalised triangulation $\mathfrak {T}$ of a 3-manifold with n tetrahedra, and a nice tree decomposition of $\mathfrak {T}$ with width k and O(n) bags, Algorithm 1 computes $\mathrm {TV}_{r}(\mathfrak {T})$ in $O(n \cdot (r-1)^{6(k+1)} \cdot k^2 \log r)$ arithmetic operations in $\mathbb {Q}(\zeta )$.

Theorem 1 shows that, if we have access to a tree decomposition of small width, then computing $\mathrm {TV}_{r}$, r fixed, becomes a linear time procedure, even for large inputs. This of course is the main benefit of fixed-parameter tractability. In our setting, however, we have an added advantage: $\mathrm {TV}_{r}$ is a topological invariant, and does not depend on our particular choice of triangulation.

Therefore, if we are faced with a large treewidth triangulation, we can retriangulate the manifold (for instance, using bistellar flips and related local moves), in an attempt to make the treewidth smaller. While this is not always possible in theory (Huszár 2018), it is extremely effective in practice, as seen in Sect. 6.

Once we have a triangulation with dual graph of small treewidth, our approach relies on our ability to actually compute explicit small-width tree decompositions efficiently. Computing the treewidth of an arbitrary graph is NP-hard (Arnborg et al. 1987), but fixed-parameter tractable with respect to the treewidth (Bodlaender 1996). Consequently, together with Theorem 1, computing the Turaev–Viro invariant $\mathrm {TV}_r(\mathfrak {T})$ of a triangulation $\mathfrak {T}$ is fixed-parameter tractable in the treewidth of the dual graph of $\mathfrak {T}$. The running time of the exact algorithm (Bodlaender 1996) is $k^{O(k^3)}n$ for treewidth-k graphs, which makes it intractable in practice. However, efficient approximation algorithms for tree decomposition exist (Amir 2010; Bodlaender et al. 2016; Feige et al. 2008). In particular, the 5-approximation algorithm of Bodlaender et al. (2016) has complexity $2^{O(k)} n$, and consequently produces tree decompositions of width reasonably close to the actual treewidth while maintaining a singly exponential complexity in the treewidth k. In particular, this yields an algorithm for the overall computation of the Turaev–Viro invariant $\mathrm {TV}_r(\mathfrak {T})$, r fixed, with singly exponential complexity in k, and a linear complexity in n, on a triangulation $\mathfrak {T}$ with n tetrahedra and treewidth k.

In the experimental section of this article, we use the polynomial time GreedyFillIn heuristic (Bodlaender and Koster 2010) which constructs tree decompositions that, in practice, are close to the actual treewidth, see Sect. 6.

Note that if—for one reason or another—we fail to obtain a small-width tree decomposition of the dual graph of our input triangulation, every tree node has $|E^*_\tau | \le \ell$, where $\ell$ is the number of edges in the triangulation. Therefore the time complexity of Algorithm 1 reduces to $O(n \cdot (r-1)^{\ell } \cdot \ell ^2 \log r)$, which is only a little slower than the backtracking algorithm (Lemma 1). This is in sharp contrast to many FPT algorithms from the literature, which—although fast for small parameters—suffer from extremely poor performance when the parameter becomes large.

5 Generalisation to the Turaev–Viro–Barrett–Westbury model

The FPT algorithm described in Sect. 4, generalises to every 3-manifold invariant which can be defined using a triangulation $\mathfrak {T}$ of the manifold and a state-sum over “local data” on $\mathfrak {T}$, that is, colourings of the edges of $\mathfrak {T}$, of the form of Eqs. (1) and (2)—for an arbitrary set of weights $|\cdot |_\theta$.

In this section we describe a very large family of 3-manifold invariants satisfying this condition, and due to Barrett and Westbury (1996). More precisely, the Turaev–Viro–Barrett–Westbury model defines a 3-manifold invariant for every (finite semisimple) spherical category, and substantially generalises the Turaev–Viro construction given in Sect. 2.1.

Roughly speaking, a spherical category is a category which admits the notion of local data (edge colourings)—stored in triangles and tetrahedra of the triangulation—from which a state-sum can be formed. The value of the state sum can then be shown to be invariant under bistellar flips on the triangulation. This is done by using a generalised version of what is called the Biedenharn–Elliott identity (Turaev 2010, Theorem VI.5.4.1). In particular this proves that the state sum defines a topological invariant.

For our purposes, the following properties of a spherical category $\mathcal {C}$ are important (we refer to Barrett and Westbury 1996 for a precise definition of a spherical category):

• $\mathcal {C}$ is equipped with a tensor product of objects $(a,b) \mapsto a \otimes b$ and of morphisms $(f,g) \mapsto f \otimes g$;

• $\mathcal {C}$ admits a trace function $\mathrm {tr}: \mathrm {Hom}(a,a) \rightarrow \mathbb {F}$, taking values in a fixed field $\mathbb {F}$, for every object a. Given two morphisms $f: a \rightarrow a$ and $g: b \rightarrow b$, the trace function satisfies $\mathrm {tr}(f \otimes g) = \mathrm {tr}(f)\mathrm {tr}(g)$, where the multiplication is in $\mathbb {F}$;

• $\mathcal {C}$ admits a finite set of nonequivalent non-zero simple objects J; and

• every $\mathrm {Hom}$ set is a finitely generated abelian group, and $\mathcal {C}$ defines an evaluation map $\mathrm {ev}: \mathrm {Hom}(a,b) \otimes \mathrm {Hom}(b,a) \rightarrow \mathbb {F}$ by $\mathrm {ev}(f \otimes g) = \mathrm {tr}(f \circ g)$.

A topological invariant is formed by taking a state-sum over all colourings of the edges E of the triangulation $\mathfrak {T}$ by simple objects of J. For each such colouring $\theta : E \rightarrow J$ we associate a value $Z(\mathfrak {T}, \theta ) \in \mathbb {F}$, computed in the following fashion. Fix an orientation of $\mathfrak {T}$, and let T be a tetrahedron of the triangulation whose triangular faces are $t_1, t_2, t_3$ and $t_4$. Assign to T the space $V(T,\theta ) = \bigotimes _{i = 1 \ldots 4} H(t_i,\theta )$, where, for a triangle $t_i$ coloured $(a,b,c) \in J^3$ by $\theta$, $H(t_i, \theta )$ is either $\mathrm {Hom}(b,a \otimes c)$ or $\mathrm {Hom}^*(b,a \otimes c) \cong \mathrm {Hom}(a \otimes c,b)$, depending on whether the orientations of (a, b, c) and T agree or not. Define the space $V(\mathfrak {T},\theta )$ to be the non-ordered tensor product of the $V(T,\theta )$ taken over all tetrahedra T of $\mathfrak {T}$. Finally, every coloured tetrahedron $T \in \mathfrak {T}$ is assigned an element $Z(T, \theta ) \in V(T,\theta )$, called a 6j-symbol.

Note that, because every triangle is shared by exactly two tetrahedra with opposite orientations, every triangle coloured (a, b, c) induces the appearance of spaces $\mathrm {Hom}(b,a\otimes c)$ and $\mathrm {Hom}(a \otimes c, b)$ in the tensor product $V(\mathfrak {T},\theta )$. The state-sum is then defined by

$$\begin{aligned} Z(\mathfrak {T}) = K^{-\#V} \sum _{\theta : E \rightarrow J} Z(\mathfrak {T},\theta ) \prod _{e \in E} \dim _q(\theta (e)) \end{aligned}$$

(3)

where

• K is a constant depending solely on the category $\mathcal {C}$, called the dimension of $\mathcal {C}$;

• $\dim _q(a)$ is the quantum dimension of the object a, defined by $\mathrm {tr}(1_a)$;

• $Z(\mathfrak {T},\theta ) := \mathrm {ev}( \bigotimes _T Z(T,\theta ))$ is the evaluation of $V(\mathfrak {T},\theta )$, i.e., the product (in $\mathbb {F}$) over all triangles of $\mathfrak {T}$ of the evaluation map on $\mathrm {Hom}(b,a\otimes c) \otimes \mathrm {Hom}(a \otimes c, b) \rightarrow \mathbb {F}$ applied to the element given by the tensor product of the 6j-symbols.

As an example, we recover the Turaev–Viro invariant $\mathrm {TV}_r$ from this construction. Using the notations of Sect. 2.1, the simple objects of J are labelled by half integers from 0 to $(r-2)/2$, and every state space $H(t,\theta )$ of a triangle whose edges are coloured with objects $a,b,c \in J$ is either the space 0, if a, b and c form a non admissible triple (in the sense of Sect. 2.1), or $\mathbb {C}$ otherwise. In Eq. (1), the product of vertex weights $|v|_\theta$ gives $K^{-\# V}$ in Eq. (3) (vertex weights are independent of $\theta$). The product of edge weights $|e|_\theta$ yields the product of $\dim _q(e)$ (edge weights are independent of the triangulation).

The dynamic programming approach of the fixed-parameter tractable algorithm of Sect. 4 is independent of the weight system in the state-sum formula of the invariant. Consequently, it adapts directly to the general definition of Turaev–Viro–Barrett–Westbury invariants.

Corollary 2

Given a generalised triangulation $\mathfrak {T}$ of a 3-manifold with n tetrahedra, a nice tree decomposition of $\mathfrak {T}$ of width k and O(n) bags, and a finite semisimple spherical category $\mathcal {C}$, there exists a fixed-parameter tractable algorithm with respect to k to compute the $\mathcal {C}$-Turaev–Viro–Barrett–Westbury invariant of $\mathfrak {T}$ in $O(n \cdot (r-1)^{6(k+1)} \cdot k^2 \log r)$ arithmetic operations in the field $\mathbb {F}$ of $\mathcal {C}$, provided $\mathcal {C}$ admits r distinct inequivalent simple objects, and all weights in Eq. 3 are precomputed.

6 Implementation and experimentation

In this section we describe an implementation of Algorithm 1 (the fixed-parameter tractable algorithm), and compare it to the backtracking algorithm via exhaustive experimentation.

The FPT algorithm is implemented in the open-source software package Regina (Burton et al. 1999): the source code is available from Regina’s public git repository. For consistency we compare it to Regina’s long-standing implementation of the backtracking algorithm.²

In our implementation, we do not compute treewidths precisely (an NP-complete problem)—instead, we implement the quadratic-time GreedyFillIn heuristic (Bodlaender and Koster 2010), which is reported to produce small widths in practice (van Dijk et al. 2006). This way, costs of building tree decompositions are insignificant (but included in the running times). For both algorithms, we use relatively naïve implementations of arithmetic in cyclotomic fields—these are asymptotically slower than described in Sect. 3, but have very small constants.

We use two data sets for our experiments, both taken from large census databases of 3-manifolds to ensure that the experiments are comprehensive and not cherry-picked.

The first census contains all 13,400 closed prime orientable manifolds that can be formed from $n \le 11$ tetrahedra (Burton 2011; Matveev 2003). This simulates “real-world” computation—the Turaev–Viro invariants were used to build this census. Since the census includes all minimal triangulations of these manifolds, we choose the representative whose heuristic tree decomposition has smallest width (since we are allowed to retriangulate).

The second data set contains some of the 11,031 triangulations from the (much larger) Hodgson–Weeks census of closed hyperbolic manifolds (Hodgson and Weeks 1994). This shows performance on larger triangulations, with n ranging from 9 to 20.

Figures 3 and 4 compare the performance of both algorithms for each data set. All running times are for $\mathrm {TV}_{7}$ at $\zeta = e^{i \pi / 7}$ (the largest r for which the experiments were feasible), and are measured on a single 3 GHz Intel Core i7 CPU. Both plots use a log-log scale with one data point per input triangulation. The results are striking: the FPT algorithm runs faster in over $99\%$ of cases, including most of the cases with largest treewidth. In the worst example the FPT algorithm runs $3.7\times$ slower than the backtracking, but both data sets have examples that run $>440\times$ faster. It is also pleasing to see a clear impact of the treewidth on the performance of the FPT algorithm, as one would expect.

Fig. 3

Running times for the 13,400 closed census manifolds. The term “width k” (an upper bound for the actual treewidth) denotes the width of the tree decomposition for the triangulation obtained from GreedyFillIn heuristic, and used for the FPT algorithm

Fig. 4

Running times for the first 500 Hodgson–Weeks census manifolds. The term “width k” (an upper bound for the actual treewidth) indicates the width of the tree decomposition for the triangulation obtained from GreedyFillIn heuristic, and used for the FPT algorithm

7 An alternate geometric interpretation

In this section, we give a geometric interpretation of admissible colourings on a triangulation of a 3-manifold $\mathfrak {T}$ in terms of normal arcs, i.e., straight lines in the interior of a triangle which are pairwise disjoint and meet the edges of a triangle, but not its vertices (see Fig. 5). More precisely, we have the following

Theorem 2

Given a 3-manifold triangulation $\mathfrak {T}$, and $r \ge 3$, an admissible colouring of the edges of $\mathfrak {T}$ with $r-1$ colours corresponds to a system of normal arcs in the 2-skeleton with $\le r -2$ arcs per triangle forming a collection of cycles on the boundary of each tetrahedron of $\mathfrak {T}$.

Proof

Following the definition of an admissible colouring from Sect. 2.1, the colours of the edges $e_1$, $e_2$, $e_3$ of a triangle f of $\mathfrak {T}$ must satisfy the parity condition and the triangle inequalities.

For a colouring $\theta (e)$ of an edge e, we define $\phi (e) = 2\theta (e)$ which is an integer; we also use the term “colouring” for $\phi$. We interpret the colourings $\phi (e_1),\phi (e_2),\phi (e_3) \in \{0, 1, \ldots , r-2\}$ as the number of intersections of normal arcs with the respective edges of the triangulation (see Fig. 5). Without loss of generality, let $\phi (e_1) \ge \phi (e_2) \ge \phi (e_3)$. We construct a system of normal arcs by first drawing $\phi (e_2)$ arcs between edge $e_1$ and $e_3$ and $\phi (e_1) - \phi (e_2)$ arcs between edge $e_1$ and $e_3$. This is always possible since $\phi (e_1) \le \phi (e_2) + \phi (e_3)$ by the triangle inequality. Furthermore, the parity condition ensures that an even number of unmatched intersections remains which, by construction, all have to be on edge $e_3$. If this number is zero we are done. Otherwise we start replacing normal arcs between $e_1$ and $e_2$ by pairs of normal arcs, one between $e_1$ and $e_3$ and one between $e_2$ and $e_3$ (see Fig. 5). In each step, the number of unmatched intersection points decreases by two. By the assumption $\phi (e_2) \ge \phi (e_3)$, this yields a system of normal arcs in f which leaves no intersection on the boundary edges unmatched. This system of normal arcs is unique for each admissible triple of colours. By the upper bound constraint, we get at most $r-2$ normal arcs on f.

Looking at the boundary of a tetrahedron t of $\mathfrak {T}$ these normal arcs form a collection of closed cycles. To see this, note that each intersection point of a normal arc in a triangle with an edge is part of exactly one normal arc in that triangle and that there are exactly two triangles sharing a given edge.

Fig. 5

Constructing a system of normal arcs from edge colourings

Now, let $\mathfrak {T}$ be a closed n-tetrahedron 3-manifold triangulation, t a tetrahedron of $\mathfrak {T}$, $f_1$ and $f_2$ two triangles of t with common edge e of colour $\phi (e)$, and $a_i$ and $b_i$ the respective non-negative numbers of the two normal arc types in $f_i$ meeting e, $i \in \{1,2\}$. Since the system of normal arcs on t forms a collection of cycles on the boundary of t, we must have $a_1 + b_1 = a_2 + b_2 \le r-2$, giving rise to a total of 6n linear equations and 12n linear inequalities on 6n variables which all admissible colourings on $\mathfrak {T}$ must satisfy. Thus, finding admissible colourings on $\mathfrak {T}$ translates to the enumeration of integer lattice points within the polytope defined by the above equalities and inequalities.

Now, if we drop the upper bound constraint above, we get a cone. Computing the Hilbert basis of integer lattice points of this cone yields a finite description of all admissible colourings for any $r \ge 3$ and, thus, the essential information to compute $\mathrm {TV}_{r} (\mathfrak {T})$ for arbitrary r. Transforming this approach into a practical algorithm is work in progress.