Tables of Spines and 3-Manifolds up to Complexity 7

Sergei V. Matveev^{1}

This paper is a development of [4]. First, I reproduce from [4] the tables of minimal special spines of closed orientable irreducible 3-manifolds up to complexity 6. The reason is that the figures in [4] were prepared in the PCX format and occupy a lot of memory (more than 500 Megabytes in compressed form). This makes it difficult downloading the paper.* Now, after redrawing them in the EPS format, they take only 5 Megabytes. I thank E. Sbrodova for the help in preparing new figures.

Second, we extend the table by adding a part of M. Ovchinnikov’s table of closed orientable irreducible manifolds of complexity 7. Almost all such manifolds have minimal special spines modeled on nonclosed chains and triangles with one, two, or three tails, see Fig. 1. The structure of such manifolds for every complexity is completely described in [4]. So we only list the remaining 22 manifolds by reproducing one minimal special spine for each of them.

Third, we include M. Sokolov’s table of the Turaev-Viro invariants of order ≤ 7 for manifolds up to complexity 6.

Figure 1: A nonclosed chain and a triangle with three tails

1 Spines and complexity of 3-manifolds

We recall basic notions of the complexity theory of 3-manifolds [3].

Definition 1.1 Let M be a compact connected 3-dimensional manifold with boundary. A subpolyhedron P ⊂ M is called a spine of M if M ↘ P, that is, M collapses to P. By a spine of a closed connected 3-manifold M we mean a spine of M \ IntB^{3} where B^{3} is a 3-ball in M. A spine of a disconnected 3-manifold is the union of spines of its connected components.

1Chelyabinsk State University and Max-Planck-Institute für Mathematik (Bonn). The author is partially supported by grants E00-1.0-2.11, RFBR-02-01-01-013, and UR.04.01.033.

Figure 2: Allowable neighborhoods in a simple polyhedron

Definition 1.2 A compact polyhedron P is said to be almost simple if the link of any of its points can be embedded into a circle with three radii. This graph, which is actually the complete graph with four vertices, will be denoted by Γ_{4}.

Definition 1.3 A compact polyhedron P is called simple, if the link of each point x ∈ P is homeomorphic to one of the following 1-dimensional polyhedra:

(a) a circle (then x is called nonsingular);

(b) a circle with a diameter (then x is a triple point);

(c) a circle with three radii (then x is a true vertex).

If, in addition, the set of all nonsingular points of P consists of open 2-cells and P has at least one true vertex, then P is called special.

Typical neighborhoods of points of a simple polyhedron are shown in Fig. 2. It is known (see [1, 3]) that any homeomorphism between special spines

can be extended to a homeomorphism between the corresponding manifolds. It means that a special spine P of a 3-manifold M may serve as a presentation of M. Moreover, M can be reconstructed from a regular neighborhood N(SP) in P of the singular graph SP of P: starting from N(SP), one can easily reconstruct P by attaching 2-cells to all the circles in ∂N(SP), and then reconstruct M. If M is orientable, then N(SP) can be embedded into R^{3}. This gives us a very convenient way for presenting 3-manifolds: we simply draw pictures.

Definition 1.4 The complexity c(M) of a compact 3-manifold M is equal to k if M possesses an almost simple spine with k true vertices and has no almost simple spines with a smaller number of true vertices.

It turns out that every closed orientable irreducible 3-manifold M of complexity k > 0 has a special spine P with exactly k true vertices. Since M has no special spines with < k true vertices, we will call P minimal. There are only three closed orientable irreducible 3-manifolds of complexity k = 0: S^{3}, RP^{3}, and L(3, 1). Their minimal almost simple spines are: the point, the projective plane, and the quotient space of D^{2} by a free action on ∂D^{2} of the group Z_{3}.

2 Manifolds of complexity ≤ 6

We order closed orientable irreducible 3-manifolds of each complexity k ≤ 6 arbitrarily and write k_{i} for the manifold numbered i among manifolds of complexity k. This method of notation is borrowed from knot theory (for example, see [7]). Fortunately for us, each of those manifolds belongs to one of the following classes I – IV.

I. L_{p,q} is the lens space with parameters p, q.

II. S^{3}/G is the quotient space of S^{3} by a free linear action of a nonabelian finite group G. All such groups are known, see [5]. They are:

1. Finite cyclic groups;

2. Groups Q_{4n}, n ≥ 2;

3. Groups D_{2^{k}(2n+1)}, k ≥ 3, n ≥ 1;

4. Groups P_{24}, P_{48}, P_{120}, and P′_{8(3^{k})}, k ≥ 2;

5. Direct products of any of these groups with a cyclic group of coprime order.

Lower indices show the orders of the groups. Presentations by generators and relations, and abelian quotients of the groups (coinciding with the first homology groups of the corresponding 3-manifolds) are the following:

1. Q_{4n} = ⟨x, y| x^{2} = (xy)^{2} = y^{n}⟩; H_{1} = Z_{2} ⊕ Z_{2} if n is even, and H_{1} = Z_{4} if n is odd.

2. D_{2^{k}(2n+1)} = ⟨x, y| x^{2^{k}} = 1, y^{2n+1} = 1, xyx^{−1} = y^{−1}⟩; H_{1} = Z_{2^{k}}.

3. P_{24} = ⟨x, y | x^{2} = (xy)^{3} = y^{3}, x^{4} = 1⟩; H_{1} = Z_{3}.

4. P_{48} = ⟨x, y | x^{2} = (xy)^{3} = y^{4}, x^{4} = 1⟩; H_{1} = Z_{2}.

5. P_{120} = ⟨x, y | x^{2} = (xy)^{3} = y^{5}, x^{4} = 1⟩; H_{1} = 0.

6. P′_{8(3^{k})} = ⟨x, y, z | x^{2} = (xy)^{2} = y^{2}, zxz^{−1} = y, zyz^{−1} = xy, z^{3^{k}} = 1⟩; H_{1} = Z_{3^{k}}.

III. Let h_{A}: T → T be a self-homeomorphism of the torus T = S^{1} × S^{1} corresponding to an unimodular integer matrix A of order 2. Then T × I/A is the Stallings manifold with fiber T and monodromy map h_{A}. In other words, T × I/A is obtained from T × I by identifying its boundary tori via h_{A}.

IV. Recall that the boundary T of the orientable I-bundle K ˜×I over the Klein bottle K is a torus. Choose a coordinate system (m, l) on the torus such that m projects onto a meridian of K (i.e., a nonseparating orientationpreserving circle) and l double covers a longitude of K (an orientation-reversing circle). Let A be an integer matrix of order 2 with determinant (-1). Then K ˜×I ∪ K ˜×I/A denotes the manifold obtained by pasting together two copies of K ˜×I via h_{A}.

All closed orientable irreducible 3-manifolds of complexity ≤ 6 and the majority of manifolds of complexity 7 are Seifert fibered manifolds. We use the following notation: (F, (p_{1}, q_{1}), …, (p_{k}, q_{k})) is the orientable Seifert manifold, where F is the base surface and (p_{i}, q_{i}), 1 ≤ i ≤ k, are nonnormalized parameters of k disjoint fibers, including all exceptional ones. If F is closed, then k must be positive. By definition, (F, (p_{1}, q_{1}), …, (p_{k}, q_{k})) can be constructed as follows:

1. Let F ˜×S^{1} be a unique orientable S^{1}-bundle over F. We cut off from it k solid tori D_{i} × S^{1} ⊂ F ˜×S^{1}, where D_{i}, 1 ≤ i ≤ k are disjoint discs in F.

2. Then we glue each solid torus D_{i}×S^{1} back along a homeomorphism taking its meridian onto a curve of the type (p_{i}, q_{i}).

We do not write Seifert structures and homology groups of lens spaces, since they are well-known: L_{p,q} = (S^{2}, (q, p)) and H_{1}(L_{p,q}; Z) = Z_{p}.

Table 1: Complexity 0

\begin{matrix}c_{i} & M \\ 0_{1} & S^{3} \\ 0_{2} & RP^{3} \\ 0_{3} & L_{3,1} \end{matrix}

Table 2: Complexity 1

\begin{matrix}c_{i} & M \\ 1_{1} & L_{4,1} \\ 1_{2} & L_{5,2} \end{matrix}

Table 3: Complexity 2

\begin{matrix}c_{i} & M \\ 2_{1} & L_{5,1} \\ 2_{2} & L_{7,2} \\ 2_{3} & L_{8,3} \end{matrix}

c_{i}  M  Seifert structure  H_{1}(M; Z)
2_{4} S^{3}/Q_{8} (S^{2}, (2, 1), (2, 1)(2, −1)) Z_{2} ⊕ Z_{2}

\begin{matrix}c_{i} & M \\ 3_{1} & L_{6,1} \\ 3_{2} & L_{9,2} \\ 3_{3} & L_{10,3} \end{matrix}  \begin{matrix}c_{i} & M \\ 3_{4} & L_{11,3} \\ 3_{5} & L_{12,5} \\ 3_{6} & L_{13,5} \end{matrix}

c_{i}  M  Seifert structure  H_{1}(M; Z)
3_{7} S^{3}/Q_{12} (S^{2}, (2, 1), (2, 1), (3, −2))  Z_{4}

Table 5: Complexity 4

\begin{matrix}c_{i} & M \\ 4_{1} & L_{7,1} \\ 4_{2} & L_{11,2} \\ 4_{3} & L_{13,3} \\ 4_{4} & L_{14,3} \\ 4_{5} & L_{15,4} \end{matrix}  \begin{matrix}c_{i} & M \\ 4_{6} & L_{16,7} \\ 4_{7} & L_{17,5} \\ 4_{8} & L_{18,5} \\ 4_{9} & L_{19,7} \\ 4_{10} & L_{21,8} \end{matrix}

c_{i}  M  Seifert structure  H_{1}(M; Z)
4_{11} S^{3}/Q_{8} × Z_{3} (S^{2}, (2, 1), (2, 1), (2, 1))  Z_{2} ⊕ Z_{6}
4_{12}  S^{3}/Q_{16}  (S^{2}, (2, 1), (2, 1), (4, −3)) Z_{2} ⊕ Z_{2}
4_{13}  S^{3}/D_{24}  (S^{2}, (2, 1), (2, 1), (3, −1))  Z_{8}
4_{14}  S^{3}/P_{24}  (S^{2}, (2, 1), (3, 1), (3, −2))  Z_{3}

\begin{matrix}c_{i} & M \\ 5_{1} & L_{8,1} \\ 5_{2} & L_{13,2} \\ 5_{3} & L_{16,3} \\ 5_{4} & L_{17,3} \\ 5_{5} & L_{17,4} \\ 5_{6} & L_{19,4} \\ 5_{7} & L_{20,9} \\ 5_{8} & L_{22,5} \\ 5_{9} & L_{23,5} \\ 5_{10} & L_{23,7} \end{matrix}  \begin{matrix}c_{i} & M \\ 5_{11} & L_{24,7} \\ 5_{12} & L_{25,7} \\ 5_{13} & L_{25,9} \\ 5_{14} & L_{26,7} \\ 5_{15} & L_{27,8} \\ 5_{16} & L_{29,8} \\ 5_{17} & L_{29,12} \\ 5_{18} & L_{30,11} \\ 5_{19} & L_{31,12} \\ 5_{20} & L_{34,13} \end{matrix}

c_{i}  M  Seifert structure  H_{1}(M; Z)
5_{21} S^{3}/Q_{8} × Z_{5} (S^{2}, (2, 1), (2, 1), (2, 3)) Z_{2} ⊕ Z_{10}
5_{22} S^{3}/Q_{12} × Z_{5} (S^{2}, (2, 1), (2, 1), (3, 2))  Z_{20}
5_{23} S^{3}/Q_{16} × Z_{3} (S^{2}, (2, 1), (2, 1), (4, −1)) Z_{2} ⊕ Z_{6}
5_{24} S^{3}/Q_{20} (S^{2}, (2, 1), (2, 1), (5, −4)) Z_{4}
5_{25} S^{3}/Q_{20} × Z_{3} (S^{2}, (2, 1), (2, 1), (5, −2))  Z_{12}
5_{26}  S^{3}/D_{40}  (S^{2}, (2, 1), (2, 1), (5, −3))  Z_{8}
5_{27}  S^{3}/D_{48}  (S^{2}, (2, 1), (2, 1), (3, 1))  Z_{16}
5_{28} S^{3}/P_{24} × Z_{5} (S^{2}, (2, 1), (3, 2), (3, −1))  Z_{15}
5_{29}  S^{3}/P_{48}  (S^{2}, (2, 1), (3, 1), (4, −3))  Z_{2}
5_{30}  S^{3}/P′_{72}  (S^{2}, (2, 1), (3, 2), (3, −2))  Z_{9}
5_{31}  S^{3}/P_{120}  (S^{2}, (2, 1), (3, 1), (5, −4))  0

\begin{matrix}c_{i} & M \\ 6_{1} & L_{9,1} \\ 6_{2} & L_{15,2} \\ 6_{3} & L_{19,3} \\ 6_{4} & L_{20,3} \\ 6_{5} & L_{21,4} \\ 6_{6} & L_{23,4} \\ 6_{7} & L_{24,5} \\ 6_{8} & L_{24,11} \\ 6_{9} & L_{27,5} \end{matrix}  \begin{matrix}c_{i} & M \\ 6_{10} & L_{28,5} \\ 6_{11} & L_{29,9} \\ 6_{12} & L_{30,7} \\ 6_{13} & L_{31,7} \\ 6_{14} & L_{31,11} \\ 6_{15} & L_{32,7} \\ 6_{16} & L_{33,7} \\ 6_{17} & L_{33,10} \\ 6_{18} & L_{34,9} \end{matrix}  \begin{matrix}c_{i} & M \\ 6_{19} & L_{35,8} \\ 6_{20} & L_{36,11} \\ 6_{21} & L_{37,8} \\ 6_{22} & L_{37,10} \\ 6_{23} & L_{39,14} \\ 6_{24} & L_{39,16} \\ 6_{25} & L_{40,11} \\ 6_{26} & L_{41,11} \\ 6_{27} & L_{41,12} \end{matrix}  \begin{matrix}c_{i} & M \\ 6_{28} & L_{41,16} \\ 6_{29} & L_{43,12} \\ 6_{30} & L_{44,13} \\ 6_{31} & L_{45,19} \\ 6_{32} & L_{46,17} \\ 6_{33} & L_{47,13} \\ 6_{34} & L_{49,18} \\ 6_{35} & L_{50,19} \\ 6_{36} & L_{55,21} \end{matrix}

c_{i}  M  Seifert structure  H_{1}(M; Z)
6_{37} S^{3}/Q_{8} × Z_{7}  (S^{2}, (2, 1), (2, 1), (2, 5)) Z_{2} ⊕ Z_{14}
6_{38} S^{3}/Q_{12} × Z_{7} (S^{2}, (2, 1), (2, 1), (3, 4))  Z_{28}
6_{39} S^{3}/Q_{16} × Z_{5} (S^{2}, (2, 1), (2, 1), (4, 1)) Z_{2} ⊕ Z_{10}
6_{40} S^{3}/Q_{16} × Z_{7} (S^{2}, (2, 1), (2, 1), (4, 3)) Z_{2} ⊕ Z_{14}
6_{41} S^{3}/Q_{20} × Z_{7} (S^{2}, (2, 1), (2, 1), (5, 2))  Z_{28}
6_{42} S^{3}/Q_{24} (S^{2}, (2, 1), (2, 1), (6, −5)) Z_{2} ⊕ Z_{2}
6_{43} S^{3}/Q_{28} × Z_{3} (S^{2}, (2, 1), (2, 1), (7, −4))  Z_{12}
6_{44} S^{3}/Q_{28} × Z_{5} (S^{2}, (2, 1), (2, 1), (7, −2))  Z_{20}
6_{45} S^{3}/Q_{32} × Z_{3} (S^{2}, (2, 1), (2, 1), (8, −5)) Z_{2} ⊕ Z_{6}
6_{46} S^{3}/Q_{32} × Z_{5} (S^{2}, (2, 1), (2, 1), (8, −3)) Z_{2} ⊕ Z_{10}
6_{47}  S^{3}/D_{56}  (S^{2}, (2, 1), (2, 1), (7, −5))  Z_{8}
6_{48}  S^{3}/D_{80}  (S^{2}, (2, 1), (2, 1), (5, −1))  Z_{16}
6_{49}  S^{3}/D_{96}  (S^{2}, (2, 1), (2, 1), (3, 5))  Z_{32}
6_{50}  S^{3}/D_{112}  (S^{2}, (2, 1), (2, 1), (7, −3))  Z_{16}
6_{51}  S^{3}/D_{160}  (S^{2}, (2, 1), (2, 1), (5, 3))  Z_{32}
6_{52} S^{3}/P_{24} × Z_{7}  (S^{2}, (2, 1), (3, 1), (3, 1))  Z_{21}
6_{53} S^{3}/P_{24} × Z_{11} (S^{2}, (2, 1), (3, 2), (3, 2))  Z_{33}
6_{54} S^{3}/P_{48} × Z_{5} (S^{2}, (2, 1), (3, 2), (4, −3))  Z_{10}
6_{55} S^{3}/P_{48} × Z_{7} (S^{2}, (2, 1), (3, 1), (4, −1))  Z_{14}
6_{56} S^{3}/P_{48} × Z_{11} (S^{2}, (2, 1), (3, 2), (4, −1))  Z_{22}
6_{57} S^{3}/P_{120} × Z_{7} (S^{2}, (2, 1), (3, 1), (5, −3))  Z_{7}
6_{58} S^{3}/P_{120} × Z_{13} (S^{2}, (2, 1), (3, 1), (5, −2))  Z_{13}
6_{59} S^{3}/P_{120} × Z_{17} (S^{2}, (2, 1), (3, 2), (5, −3))  Z_{17}
6_{60} S^{3}/P_{120} × Z_{23} (S^{2}, (2, 1), (3, 2), (5, −2))  Z_{23}
6_{61} S^{3}/P′_{216} (S^{2}, (2, 1), (3, 2), (3, 1)) Z_{27}
6_{62}  (S^{2}, (3, 2), (3, 1), (3, −2)) Z_{3} ⊕ Z_{3}
6_{63}  (S^{2}, (3, 2), (3, 2), (3, −2)) Z_{3} ⊕ Z_{6}
6_{64}  (S^{2}, (3, 2), (3, 2), (3, −1)) Z_{3} ⊕ Z_{9}

8

Table 7: Complexity 6 (continued)

c_{i}  M  Seifert structure  H_{1}(M; Z)

6_{65}  T × I/ ( \begin{matrix}1 − 1 \\ 1 0 \end{matrix} )  (S^{2}, (2, −1), (3, 1), (6, 1))  Z
6_{66}  T × I/ ( \begin{matrix}0 1 \\ −1 0 \end{matrix} )  (S^{2}, (2, −1), (4, 1), (4, 1))  Z_{2} ⊕ Z
6_{67}  T × I/ ( \begin{matrix}0 1 \\ −1 − 1 \end{matrix} )  (S^{2}, (3, −1), (3, 1), (3, 1))  Z_{3} ⊕ Z
6_{68}  T × I/ ( \begin{matrix}−1 0 \\ −1 − 1 \end{matrix} )  (K, (1, 1))  Z_{4} ⊕ Z
6_{69}  T × I/ ( \begin{matrix}1 0 \\ 1 1 \end{matrix} )  (T, (1, 1))  Z ⊕ Z
6_{70}  T × I/ ( \begin{matrix}−1 0 \\ 0 − 1 \end{matrix} )  \begin{matrix}(S^{2}, (2, 1), (2, −1), (2, 1), \\ (2, −1)) \end{matrix} Z_{2} ⊕ Z_{2} ⊕ Z
6_{71}  T × I/ ( \begin{matrix}1 0 \\ 0 1 \end{matrix} )  T × S^{1}  Z ⊕ Z ⊕ Z
6_{72} K ˜×I ∪ K ˜×I/ ( \begin{matrix}−1 0 \\ −1 1 \end{matrix} )  \begin{matrix}(S^{2}, (2, 1), (2, 1), (2, 1), \\ (2, −1)) \end{matrix}  Z_{2} ⊕ Z_{2} ⊕ Z_{4}
6_{73} K ˜×I ∪ K ˜×I/ ( \begin{matrix}0 1 \\ 1 0 \end{matrix} )  (RP^{2}, (2, 1), (2, −1))  Z_{4} ⊕ Z_{4}
6_{74} K ˜×I ∪ K ˜×I/ ( \begin{matrix}1 1 \\ 1 0 \end{matrix} )  (RP^{2}, (2, 1), (2, 1))  Z_{4} ⊕ Z_{4}

3 Spines of manifolds up to complexity 6

For any manifold k_{i}, 1 ≤ k ≤ 6, we present all its minimal special spines. Manifolds 0_{1} − 0_{3} of complexity 0 are presented by their minimal almost simple spines. The spines are given by regular neighborhoods of their singular graphs.

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

4 Minimal spines of some manifolds of complexity 7

Looking at the previous table of manifolds of complexity ≤ 6, one can see that the great majority of them have minimal spines modeled on nonclosed chains and triangles with tails. The same tendency holds for manifolds of complexity 7. In order to spare place, we reproduce only a part of Ovchinnikov’s table [6] of all closed orientable irreducible 3-manifolds of complexity 7. Namely, we list below only those manifolds of complexity 7 that do not admit minimal spines of the two aforementioned types. Each manifold is represented by only one minimal spine chosen arbitrarily. We number the manifold in the same order as they appear in [6]. The star in notation 7_{k}^{∗} tells us that the numbering differs from the original one (for example, manifold 7_{1}^{∗} coincides with the manifold 7_{88} from [6]).

We present 3-manifolds as follows. Let D_{k}, k = 1, 2, be an oriented Seifert manifold of the type (D^{2}, (2, 1), (3, k)). In other words, D_{k} is obtained from D^{2} × S^{1} by cutting off two disjoint fibered solid tori and gluing them back via homeomorphisms taking their meridians onto curves of the types (2, 1) and (3, k), respectively. We equip the boundary torus of each D_{k} with the coordinate system (μ, λ), where μ is a meridian of D^{2} × S^{1} and λ is a fiber. Let A be an integer matrix of order 2 with determinant (-1). Then D_{k} ∪ K ˜×I/A denotes the manifold obtained by attaching K ˜×I to D_{k} via a homeomorphism h_{A}: ∂(K ˜×I) → ∂D_{k} corresponding to A.

Table 8: Some manifolds of complexity 7

c_{i}  M  Seifert structure  H_{1}(M; Z)
7_{1}^{∗}  S^{3}/Q_{28}  (S^{2}, (2, −1), (2, 1), (7, 1))  Z_{4}
7_{2}^{∗}  (S^{2}, (2, −1), (3, 1), (7, 1))  0
7_{3}^{∗}  (S^{2}, (2, −1), (4, 1), (4, 1))  Z_{2}
7_{4}^{∗}  T × I/ ( \begin{matrix}1 0 \\ 2 1 \end{matrix} )  (T^{2}, (1, 2))  Z ⊕ Z ⊕ Z_{2}
7_{5}^{∗}  T × I/ ( \begin{matrix}−1 0 \\ −2 − 1 \end{matrix} )  (K^{2}, (1, 2))  Z_{2} ⊕ Z_{2} ⊕ Z
7_{6}^{∗}  T × I/ ( \begin{matrix}−2 − 1 \\ −1 − 1 \end{matrix} )  −  Z ⊕ Z_{5}
7_{7}^{∗}  T × I/ ( \begin{matrix}2 1 \\ 1 1 \end{matrix} )  −  Z
7_{8}^{∗} K ˜×I ∪ K ˜×I/ ( \begin{matrix}1 0 \\ −2 − 1 \end{matrix} )  (T^{2}, (2, 1), (2, 1), (2, 1), (2, 1))  Z_{2} ⊕ Z_{2} ⊕ Z_{8}
7_{9}^{∗} K ˜×I ∪ K ˜×I/ ( \begin{matrix}−1 − 1 \\ 1 2 \end{matrix} )  −  Z_{4} ⊕ Z_{4}
7_{10}^{∗} K ˜×I ∪ K ˜×I/ ( \begin{matrix}1 − 1 \\ −2 1 \end{matrix} )  −  Z_{4} ⊕ Z_{8}
7_{11}^{∗} K ˜×I ∪ K ˜×I/ ( \begin{matrix}0 1 \\ 1 − 2 \end{matrix} )  (RP^{2}, (2, 1), (2, 1))  Z_{4} ⊕ Z_{4}
7_{12}^{∗} K ˜×I ∪ K ˜×I/ ( \begin{matrix}1 − 2 \\ −1 1 \end{matrix} )  −  Z_{2} ⊕ Z_{2} ⊕ Z_{4}
7_{13}^{∗}  D_{2} ∪ K ˜×I/ ( \begin{matrix}0 1 \\ 1 0 \end{matrix} )  (RP^{2}, (2, 1), (3, 2))  Z_{24}
7_{14}^{∗} D_{2} ∪ K ˜×I/ ( \begin{matrix}1 0 \\ −1 − 1 \end{matrix} )  (RP^{2}, (2, 1), (2, 1), (2, 1)(3, −1))  Z_{2} ⊕ Z_{14}
7_{15}^{∗} D_{1} ∪ K ˜×I/ ( \begin{matrix}1 0 \\ −1 − 1 \end{matrix} )  (S^{2}, (2, 1), (2, 1), (2, 1)(3, −2))  Z_{2} ⊕ Z_{10}
7_{16}^{∗} D_{2} ∪ K ˜×I/ ( \begin{matrix}−1 − 1 \\ 0 1 \end{matrix} )  −  Z_{4}
7_{17}^{∗} D_{1} ∪ K ˜×I/ ( \begin{matrix}−1 − 1 \\ 0 1 \end{matrix} )  −  Z_{4}
7_{18}^{∗}  D_{2} ∪ K ˜×I/ ( \begin{matrix}−1 0 \\ 0 1 \end{matrix} )  (S^{2}, (2, 1), (2, 1), (2, 1)(3, −4))  Z_{2} ⊕ Z_{2}
7_{19}^{∗}  D_{2} ∪ K ˜×I/ ( \begin{matrix}0 1 \\ 1 − 1 \end{matrix} )  (RP^{2}, (2, 1), (3, 2))  Z_{24}
7_{20}^{∗}  D_{1} ∪ K ˜×I/ ( \begin{matrix}0 1 \\ 1 − 1 \end{matrix} )  (RP^{2}, (2, 1), (3, 1))  Z_{24}
7_{21}^{∗} D_{2} ∪ K ˜×I/ ( \begin{matrix}1 − 1 \\ −1 0 \end{matrix} )  −  Z_{28}
7_{22}^{∗} D_{1} ∪ K ˜×I/ ( \begin{matrix}1 − 1 \\ −1 0 \end{matrix} )  −  Z_{20}

40 41 42 43 44

5 Tables of Turaev-Viro invariants

The tables were composed by my former student M. Sokolov. They contain the values of Turaev-Viro invariants of order ≤ 7 and their summands for all closed orientable irreducible 3-manifolds up to complexity 6. The invariants are presented by polynomials of q, where q is a primitive root of unity of degree 2r. For the sake of compactness of notation, we write σ_{k} instead of q^{k} + q^{−k}.

Table 9: TV-invariants for manifolds of complexity 0 – 2

c_{i} M ν 3  4  5  6  7
0 1  1  1  1  1
0_{1} S^{3} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  1
0 1  2  σ_{2} + 2  4  −σ_{3} + 2σ_{2} + 3
0_{2} RP^{3} 1 -1  −σ_{1}  −σ_{2} − 2  −2σ_{1}  σ_{3} − 2σ_{2} − 3
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  −2σ_{1} + 4  0
0 1  1  σ_{2} + 2  3  σ_{2} + 2
0_{3} L_{3,1} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  3  σ_{2} + 2
0 1  2  1  4  σ_{2} + 2
1_{1} L_{4,1} 1 0  0  0  0  0
2 1  0  1  0  σ_{2} + 2
Σ 2  2  2  4  2σ_{2} + 4
0 1  1  0  1  −σ_{3} + 2σ_{2} + 3
1_{2} L_{5,2} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  0  1  −σ_{3} + 2σ_{2} + 3
0 1  1  σ_{2} + 3  1  −σ_{3} + 2σ_{2} + 3
2_{1} L_{5,1} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 3  1  −σ_{3} + 2σ_{2} + 3
0 1  1  σ_{2} + 2  1  0
2_{2} L_{7,2} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  1  0
0 1  2  σ_{2} + 2  4  1
2_{3} L_{8,3} 1 0  0  0  0  0
2 1  2  σ_{2} + 2  0  1
Σ 2  4  2σ_{2} + 4  4  2
0 1  4  σ_{2} + 4  10  2σ_{2} + 7
2_{4} Q_{8} 1 0  0  0  0  0
2 3  6  3σ_{2} + 12  18  6σ_{2} + 21
Σ 4  10  4σ_{2} + 16  28  8σ_{2} + 28

Table 10: TV-invariants for manifolds of complexity 3

c_{i} M ν 3  4  5  6  7
0 1  2  1  6  1
3_{1} L_{6,1} 1 -1  σ_{1}  -1  0  -1
2 0  0  0  0  0
Σ 0 σ_{1} + 2  0  6  0
0 1  1  1  3  −σ_{3} + 2σ_{2} + 3
3_{2} L_{9,2} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  3  −σ_{3} + 2σ_{2} + 3
0 1  2  0  4  σ_{2} + 2
3_{3} L_{10,3} 1 -1  σ_{1}  0  2σ_{1}  −σ_{2} − 2
2 0  0  0  0  0
Σ 0 σ_{1} + 2  0  2σ_{1} + 4  0
0 1  1  1  1  σ_{2} + 2
3_{4} L_{11,3} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  σ_{2} + 2
0 1  2  σ_{2} + 2  6  −σ_{3} + 2σ_{2} + 3
3_{5} L_{12,5} 1 0  0  0  0  0
2 1  0  σ_{2} + 2  6  −σ_{3} + 2σ_{2} + 3
Σ 2  2  2σ_{2} + 4  12  −2σ_{3} + 4σ_{2} + 6
0 1  1  σ_{2} + 2  1  1
3_{6} L_{13,5} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  1  1
0 1  2  σ_{2} + 4  10  −σ_{3} + 2σ_{2} + 5
3_{7} Q_{12} 1 0  0  0  0  0
2 1  0  σ_{2} + 4  6  −σ_{3} + 2σ_{2} + 5
Σ 2  2  2σ_{2} + 8  16  −2σ_{3} + 4σ_{2} + 10
0 1  1  σ_{2} + 2  1  −σ_{3} + 3σ_{2} + 6
4_{1} L_{7,1} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  1  −σ_{3} + 3σ_{2} + 6
0 1  1  1  1  σ_{2} + 2
4_{2} L_{11,2} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  σ_{2} + 2

c_{i}  M  ν 3  4  5  6  7
0 1  1  σ_{2} + 2  1  1
4_{3}  L_{13,3}  1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  1  1
0 1  2  1  4  0
4_{4}  L_{14,3}  1 -1  −σ_{1}  -1  2σ_{1}  0
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  2σ_{1} + 4  0
0 1  1  σ_{2} + 3  3  1
4_{5}  L_{15,4}  1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 3  3  1
0 1  2  1  4  −σ_{3} + 2σ_{2} + 3
4_{6}  L_{16,7}  1 0  0  0  0  0
2 1  2  1  0  −σ_{3} + 2σ_{2} + 3
Σ 2  4  2  4  −2σ_{3} + 4σ_{2} + 6
0 1  1  σ_{2} + 2  1  σ_{2} + 2
4_{7}  L_{17,5}  1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  1  σ_{2} + 2
0 1  2  σ_{2} + 2  6  σ_{2} + 2
4_{8}  L_{18,5}  1 -1  −σ_{1}  −σ_{2} − 2  0  −σ_{2} − 2
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  6  0
0 1  1  1  1  −σ_{3} + 2σ_{2} + 3
4_{9}  L_{19,7}  1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  −σ_{3} + 2σ_{2} + 3
0 1  1  1  3  −σ_{3} + 3σ_{2} + 6
4_{10} L_{21,8}  1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  3  −σ_{3} + 3σ_{2} + 6
0 1  4  2σ_{2} + 5  12  −2σ_{3} + 7σ_{2} + 12
4_{11} Q_{8} × Z_{3} 1 0  0  0  0  0
2 3  6  6σ_{2} + 15  36  −6σ_{3} + 21σ_{2} + 36
Σ 4  10  8σ_{2} + 20  48  −8σ_{3} + 28σ_{2} + 48
0 1  4  1  10  −σ_{3} + 2σ_{2} + 5
4_{12}  Q_{16}  1 -2  0  -2  4σ_{1}  2σ_{3} − 4σ_{2} − 10
2 1  2  1  6  −σ_{3} + 2σ_{2} + 5
Σ 0  6  0  4σ_{1} + 16  0

c_{i}  M ν 3 4  5  6  7
0 1 2 2σ_{2} + 5 10 −2σ_{3} + 5σ_{2} + 8
4_{13} D_{24} 1 0 0 0 0 0
2 1 2 2σ_{2} + 5 6 −2σ_{3} + 5σ_{2} + 8
Σ 2 4 4σ_{2} + 10 16 −4σ_{3} + 10σ_{2} + 16
0 1 1 2σ_{2} + 5 3 −2σ_{3} + 4σ_{2} + 6
4_{14} P_{24} 1 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1 2σ_{2} + 5 3 −2σ_{3} + 4σ_{2} + 6
0 1 2 σ_{2} + 2 4  1
5_{1} L_{8,1} 1 0 0 0 0 0
2 1 -2 σ_{2} + 2 0  1
Σ 2 0 2σ_{2} + 4 4  2
0 1 1 σ_{2} + 2 1  1
5_{2} L_{13,2} 1 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1 σ_{2} + 2 1  1
0 1 2  1  4  −σ_{3} + 2σ_{2} + 3
5_{3} L_{16,3} 1 0 0  0  0  0
2 1 -2  1  0  −σ_{3} + 2σ_{2} + 3
Σ 2 0  2  4 −2σ_{3} + 4σ_{2} + 6
0 1 1 σ_{2} + 2 1  σ_{2} + 2
5_{4} L_{17,3} 1 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1 σ_{2} + 2 1  σ_{2} + 2
0 1 1 σ_{2} + 2 1  σ_{2} + 2
5_{5} L_{17,4} 1 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1 σ_{2} + 2 1  σ_{2} + 2
0 1 1  1  1  −σ_{3} + 2σ_{2} + 3
5_{6} L_{19,4} 1 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1  1  1  −σ_{3} + 2σ_{2} + 3
0 1 2 σ_{2} + 3 4  1
5_{7} L_{20,9} 1 0 0 0 0 0
2 1 0 σ_{2} + 3 0  1
Σ 2 2 2σ_{2} + 6 4  2

c_{i}  M ν 3  4  5  6  7
0 1  2  σ_{2} + 2  4  1
5_{8} L_{22,5} 1 -1  σ_{1}  −σ_{2} − 2  −2σ_{1}  -1
2 0  0  0  0  0
Σ 0 σ_{1} + 2  0  −2σ_{2} + 4  0
0 1  1  σ_{2} + 2  1  −σ_{3} + 2σ_{2} + 3
5_{9} L_{23,5} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  1  −σ_{3} + 2σ_{2} + 3
0 1  1  σ_{2} + 2  1  −σ_{3} + 2σ_{2} + 3
5_{10} L_{23,7} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  1  −σ_{3} + 2σ_{2} + 3
0 1  2  1  6  σ_{2} + 2
5_{11} L_{24,7} 1 0  0  0  0  0
2 1  -2  1  -6  σ_{2} + 2
Σ 2  0  2  0  2σ_{2} + 4
0 1  1  0  1  σ_{2} + 2
5_{12} L_{25,7} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  0  1  σ_{2} + 2
0 1  1  σ_{2} + 3  1  σ_{2} + 2
5_{13} L_{25,9} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 3  1  σ_{2} + 2
0 1  2  1  4  −σ_{3} + 2σ_{2} + 3
5_{14} L_{26,7} 1 -1  σ_{1}  -1  −2σ_{1}  σ_{3} − 2σ_{2} − 3
2 0  0  0  0  0
Σ 0 σ_{1} + 2  0  −2σ_{1} + 4  0
0 1  1  σ_{2} + 2  3  1
5_{15} L_{27,8} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  3  1
0 1  1  1  1  1
5_{16} L_{29,8} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  1

Table 14: TV-invariants for manifolds of complexity 5 (continued)

c_{i}  M  ν 3  4  5  6  7
0 1  1  1  1  1
5_{17} L_{29,12} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  1
0 1  2  σ_{2} + 3  6  −σ_{3} + 2σ_{2} + 3
5_{18} L_{30,11} 1 -1  −σ_{1}  −σ_{2} − 3  0  σ_{3} − 2σ_{2} − 3
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  6  0
0 1  1  1  1  σ_{2} + 2
5_{19} L_{31,12} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  σ_{2} + 2
0 1  2  1  4  1
5_{20} L_{34,13} 1 -1  −σ_{1}  -1  2σ_{1}  -1
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  2σ_{1} + 4  0
5_{21} Q_{8} × Z_{5}
0 1  4  σ_{2} + 3  10  −3σ_{3} + 6σ_{2} + 11
1 0  0  0  0  0
2 3  6  3σ_{2} + 9  18  −9σ_{3} + 18σ_{2} + 33
Σ 4  10  4σ_{2} + 12  28  −12σ_{3} + 24σ_{2} + 44
5_{22} Q_{12} × Z_{5}
0 1  2  σ_{2} + 3  10  −2σ_{3} + 5σ_{2} + 8
1 0  0  0  0  0
2 1  0  σ_{2} + 3  6  −2σ_{3} + 5σ_{2} + 8
Σ 2  2  2σ_{2} + 6  16  −4σ_{3} + 10σ_{2} + 16
5_{23} Q_{16} × Z_{3}
0 1  4  σ_{2} + 2  12  2σ_{2} + 5
1 -2  0  −2σ_{2} − 4  0  −4σ_{2} − 10
2 1  2  σ_{2} + 2  12  2σ_{2} + 5
Σ 0  6  0  24  0

c_{i}  M  ν 3  4  5  6  7
0 1  2  3σ_{2} + 5 4  2σ_{2} + 7
5_{24}  Q_{20}  1 0  0  0  0  0
2 1  0  3σ_{2} + 5 0  2σ_{2} + 7
Σ 2  2  6σ_{2} + 10 4  4σ_{2} + 14
0 1  2  −σ_{2} + 1 6 −2σ_{3} + 7σ_{2} + 12
5_{25} Q_{20} × Z_{3} 1 0  0  0  0  0
2 1  0  −σ_{2} + 1 6 −2σ_{3} + 7σ_{2} + 12
Σ 2  2  −2σ_{2} + 2 12 −4σ_{3} + 14σ_{2} + 24
0 1  2  −σ_{2} + 1 4 −3σ_{3} + 6σ_{2} + 11
5_{26}  D_{40}  1 0  0  0  0  0
2 1  2  −σ_{2} + 1 0 −3σ_{3} + 6σ_{2} + 11
Σ 2  4  −2σ_{2} + 2 4 −6σ_{3} + 12σ_{2} + 22
0 1  2  σ_{2} + 4 10  2σ_{2} + 5
5_{27}  D_{48}  1 0  0  0  0  0
2 1  2  σ_{2} + 4  6  2σ_{2} + 5
Σ 2  4  2σ_{2} + 8 16  4σ_{2} + 10
0 1  1  σ_{2} + 3  3  2σ_{2} + 4
5_{28} P_{24} × Z_{5} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 3  3  2σ_{2} + 4
0 1  2  σ_{2} + 2 10  2
5_{29}  P_{48}  1 -1  σ_{1}  −σ_{2} − 2 0  -2
2 0  0  0  0  0
Σ 0 σ_{1} + 2  0  10  0
0 1  1  σ_{2} + 4  3  2
5_{30}  P′_{72}  1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 4  3  2
0 1  1  3σ_{2} + 5 1 −2σ_{3} + 5σ_{2} + 8
5_{31}  P_{120}  1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  3σ_{2} + 5 1 −2σ_{3} + 5σ_{2} + 8

c_{i}  M  ν 3 4  5  6  7
0 1 1  1  3 −σ_{3} + 2σ_{2} + 3
6_{1} L_{9,1} 1 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1  1  3 −σ_{3} + 2σ_{2} + 3
0 1 1  0  3  1
6_{2} L_{15,2} 1 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1  0  3  1
0 1 1  1  1 −σ_{3} + 2σ_{2} + 3
6_{3} L_{19,3} 1 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1  1  1 −σ_{3} + 2σ_{2} + 3
0 1 2  0  4  1
6_{4} L_{20,3} 1 0 0  0  0  0
2 1 0  0  0  1
Σ 2 2  0  4  2
0 1 1  1  3  0
6_{5} L_{21,4} 1 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1  1  3  0
0 1 1 σ_{2} + 2 1 −σ_{3} + 2σ_{2} + 3
6_{6} L_{23,4} 1 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1 σ_{2} + 2 1 −σ_{3} + 2σ_{2} + 3
0 1 2  1  6  σ_{2} + 2
6_{7} L_{24,5} 1 0 0  0  0  0
2 1 2  1  -6  σ_{2} + 2
Σ 2 4  2  0  2σ_{2} + 4
0 1 2  1  6  σ_{2} + 2
6_{8} L_{24,11} 1 0 0  0  0  0
2 1 2  1  6  σ_{2} + 2
Σ 2 4  2  12  2σ_{2} + 4
0 1 1 σ_{2} + 2 3  1
6_{9} L_{27,5} 2 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1 σ_{2} + 2 3  1

c_{i}  M  ν 3  4  5  6  7
0 1  2  σ_{2} + 2 4  0
6_{10} L_{28,5} 1 0  0  0  0  0
2 1  0  σ_{2} + 2 0  0
Σ 2  2  2σ_{2} + 4 4  0
0 1  1  1  1  1
6_{11} L_{29,9} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  1
0 1  2  0  6 −σ_{3} + 2σ_{2} + 3
6_{12} L_{30,7} 1 -1  −σ_{1}  0  0 σ_{3} − 2σ_{2} − 3
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  6  0
0 1  1  1  1  σ_{2} + 2
6_{13} L_{31,7} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  σ_{2} + 2
0 1  1  1  1  σ_{2} + 2
6_{14} L_{31,11} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  σ_{2} + 2
0 1  2  σ_{2} + 2 4  σ_{2} + 2
6_{15} L_{32,7} 1 0  0  0  0  0
2 1  2  σ_{2} + 2 0  σ_{2} + 2
Σ 2  4  2σ_{2} + 4 4  2σ_{2} + 4
0 1  1  σ_{2} + 2 3 −σ_{3} + 2σ_{2} + 3
6_{16} L_{33,7} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2 3 −σ_{3} + 2σ_{2} + 3
0 1  1  σ_{2} + 2 3 −σ_{3} + 2σ_{2} + 3
6_{17} L_{33,10} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2 3 −σ_{3} + 2σ_{2} + 3

c_{i}  M  ν 3  4  5  6  7
0 1  2  1  4  1
6_{18} L_{34,9} 1 -1  −σ_{1}  -1  2σ_{2}  -1
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  2σ_{1} + 4  0
0 1  1  0  1  −σ_{3} + 3σ_{2} + 6
6_{19} L_{35,8} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  0  1  −σ_{3} + 3σ_{2} + 6
0 1  2  1  6  1
6_{20} L_{36,11} 1 0  0  0  0  0
2 1  0  1  -6  1
Σ 2  2  2  0  2
0 1  1  σ_{2} + 2  1  −σ_{3} + 2σ_{2} + 3
6_{21} L_{37,8} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  1  −σ_{3} + 2σ_{2} + 3
0 1  1  σ_{2} + 2  1  −σ_{3} + 2σ_{2} + 3
6_{22} L_{37,10} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  1  −σ_{3} + 2σ_{2} + 3
0 1  1  1  3  σ_{2} + 2
6_{23} L_{39,14} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  3  σ_{2} + 2
0 1  1  1  3  σ_{2} + 2
6_{24} L_{39,16} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  3  σ_{2} + 2
0 1  2  σ_{2} + 3  4  −σ_{3} + 2σ_{2} + 3
6_{25} L_{40,11} 1 0  0  0  0  0
2 1  2  σ_{2} + 3  0  −σ_{3} + 2σ_{2} + 3
Σ 2  4  2σ_{2} + 6  4  −2σ_{3} + 4σ_{2} + 6
0 1  1  1  1  1
6_{26} L_{41,11} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  1

c_{i}  M  ν 3  4  5  6  7
0 1  1  1  1  1
6_{27} L_{41,12} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  1
0 1  1  1  1  1
6_{28} L_{41,16} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  1
0 1  1  σ_{2} + 2  1  1
6_{29} L_{43,12} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  1  1
0 1  2  1  4  −σ_{3} + 2σ_{2} + 3
6_{30} L_{44,13} 1 0  0  0  0  0
2 1  0  1  0  −σ_{3} + 2σ_{2} + 3
Σ 2  2  2  4  −2σ_{3} + 4σ_{2} + 6
0 1  1  σ_{2} + 3  3  σ_{2} + 2
6_{31} L_{45,19} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 3  3  σ_{2} + 2
0 1  2  1  4  σ_{2} + 2
6_{32} L_{46,17} 1 -1  −σ_{1}  -1  −2σ_{1}  −σ_{2} − 2
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  −2σ_{1} + 4  0
0 1  1  σ_{2} + 2  1  −σ_{3} + 2σ_{2} + 3
6_{33} L_{47,13} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 2  1  −σ_{3} + 2σ_{2} + 3
0 1  1  1  1  0
6_{34} L_{49,18} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  1  1  0
0 1  2  σ_{2} + 3  4  1
6_{35} L_{50,19} 1 -1  −σ_{1}  −σ_{2} − 3  −2σ_{1}  -1
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  −2σ_{1} + 4  0

c_{i}  M  ν 3 4  5  6  7
0 1 1  σ_{2} + 3  1  1
6_{36} L_{55,21} 1 0 0  0  0  0
2 0 0  0  0  0
Σ 1 1  σ_{2} + 3  1  1
6_{37} Q_{8} × Z_{7}
0 1 4 2σ_{2} + 5  10  −σ_{3} + 3σ_{2} + 6
1 0 0  0  0  0
2 3 6 6σ_{2} + 15  18  −3σ_{3} + 9σ_{2} + 18
Σ 4 10 8σ_{2} + 20  28  −4σ_{3} + 12σ_{2} + 24
6_{38} Q_{12} × Z_{7}
0 1 2 2σ_{2} + 5  10  −σ_{3} + 3σ_{2} + 6
1 0 0  0  0  0
2 1 0 2σ_{2} + 5  6  −σ_{3} + 3σ_{2} + 6
Σ 2 2 4σ_{2} + 10  16  −2σ_{3} + 6σ_{2} + 12
6_{39} Q_{16} × Z_{5}
0 1 4  σ_{2} + 3  10  −2σ_{3} + 5σ_{2} + 8
1 -2 0 −2σ_{2} − 6  −4σ_{1}  4σ_{3} − 10σ_{2} − 16
2 1 2  σ_{2} + 3  6  −2σ_{3} + 5σ_{2} + 8
Σ 0 6  0  −4σ_{1} + 16  0
6_{40} Q_{16} × Z_{7}
0 1 4  σ_{2} + 2  10  −σ_{3} + 3σ_{2} + 6
1 -2 0 −2σ_{2} − 4  −4σ_{1}  2σ_{3} − 6σ_{2} − 12
2 1 2  σ_{2} + 2  6  −σ_{3} + 3σ_{2} + 6
Σ 0 6  0  −4σ_{1} + 16  0
6_{41} Q_{20} × Z_{7}
0 1 2 −σ_{2} + 1  4  −σ_{3} + 3σ_{2} + 6
1 0 0  0  0  0
2 1 0 −σ_{2} + 1  0  −σ_{3} + 3σ_{2} + 6
Σ 2 2 −2σ_{2} + 2  4  −2σ_{3} + 6σ_{2} + 12

c_{i} M ν 3  4  5  6  7
0 1  4  1  16  σ_{2} + 2
6_{42} Q_{24} 1 0  0  0  0  0
2 3  -2  3  12  3σ_{+}6
Σ 4  2  4  28  4σ_{2} + 8
6_{43} Q_{28} × Z_{3}
0 1  2  2σ_{2} + 5  6  −σ_{3} − σ_{2} + 2
1 0  0  0  0  0
2 1  0  2σ_{2} + 5  6  −σ_{3} − σ_{2} + 2
Σ 2  2  4σ_{2} + 10  12  −2σ_{3} − 2σ_{2} + 4
6_{44} Q_{28} × Z_{5}
0 1  2  σ_{2} + 3  4  3σ_{3} − σ_{2} + 2
1 0  0  0  0  0
2 1  0  σ_{2} + 3  0  3σ_{3} − σ_{2} + 2
Σ 2  2  2σ_{2} + 6  4  6σ_{3} − 2σ_{2} + 4
6_{45} Q_{32} × Z_{3}
0 1  4  2σ_{2} + 5  12  −σ_{3} + 2σ_{2} + 3
1 -2  −4σ_{1}  −4σ_{2} − 10  0  2σ_{3} − 4σ_{2} − 6
2 1  2  2σ_{2} + 5  12  −σ_{3} + 2σ_{2} + 3
Σ 0 −4σ_{1} + 6  0  24  0
6_{46} Q_{32} × Z_{5}
0 1  4  σ_{2} + 3  10  1
1 -2  −4σ_{1}  −2σ_{2} − 6  4σ_{1}  -2
2 1  2  σ_{2} + 3  6  1
Σ 0 −4σ_{1} + 6  0  4σ_{1} + 16  0
0 1  2  2σ_{2} + 5  4  3σ_{3} − σ_{2} + 2
6_{47} D_{56} 1 0  0  0  0  0
2 1  2  2σ_{2} + 5  0  3σ_{3} − σ_{2} + 2
Σ 2  4  4σ_{2} + 10  4  6σ_{3} − 2σ_{2} + 4

c_{i}  M ν 3  4  5  6  7
0 1  2  3σ_{2} + 5 4 −2σ_{3} + 7σ_{2} + 12
6_{48} D_{80} 1 0  0  0  0  0
2 1  2  3σ_{2} + 5 0 −2σ_{3} + 7σ_{2} + 12
Σ 2  4  6σ_{2} + 10 4 −4σ_{3} + 14σ_{2} + 24
0 1  2  2σ_{2} + 5 10 −σ_{3} + 2σ_{2} + 5
6_{49} D_{96} 1 0  0  0  0  0
2 1  2  2σ_{2} + 5 6  −σ_{3} + 2σ_{2} + 5
Σ 2  4  4σ_{2} + 10 16 −2σ_{3} + 4σ_{2} + 10
0 1  2  σ_{2} + 4  4  −σ_{3} − σ_{2} + 2
6_{50} D_{112} 1 0  0  0  0  0
2 1  2  σ_{2} + 4  0  −σ_{3} − σ_{2} + 2
Σ 2  4  2σ_{2} + 8 4 −2σ_{3} − 2σ_{2} + 4
0 1  2  −σ_{2} + 1 4  2σ_{2} + 7
6_{51} D_{160} 1 0  0  0  0  0
2 1  2  −σ_{2} + 1 0  2σ_{2} + 7
Σ 2  4  −2σ_{1} + 2 4  4σ_{2} + 14
6_{52} P_{24} × Z_{7}
0 1  1  σ_{2} + 4  3  −σ_{3} + 3σ_{2} + 6
1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 4  3  −σ_{3} + 3σ_{2} + 6
6_{53} P_{24} × Z_{11}
0 1  1  2σ_{2} + 5 3  2
1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  2σ_{2} + 5 3  2
6_{54} P_{48} × Z_{5}
0 1  2  σ_{2} + 3 10 −2σ_{3} + 4σ_{2} + 6
1 -1  −σ_{1}  −σ_{2} − 3 0  2σ_{3} − 4σ_{2} − 6
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  10  0

c_{i} M ν 3  4  5  6  7

6_{55} P_{48} × Z_{7}
0 1  2  1  10 −σ_{3} + 3σ_{2} + 6
1 -1  σ_{1}  -1  0  σ_{3} − 3σ_{2} − 6
2 0  0  0  0  0
Σ 0 σ_{1} + 2  0  10  0
6_{56} P_{48} × Z_{11}
0 1  2  σ_{2} + 2 10  2σ_{2} + 4
1 -1  −σ_{1}  −σ_{2} − 2 0  −2σ_{2} − 4
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  10  0
6_{57} P_{120} × Z_{7}
0 1  1  −σ_{2} + 1 1 −σ_{3} + 3σ_{2} + 6
1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  −σ_{2} + 1 1 −σ_{3} + 3σ_{2} + 6
6_{58} P_{120} × Z_{13}
0 1  1  −σ_{2} + 1 1 −2σ_{3} + 5σ_{2} + 8
1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  −σ_{2} + 1 1 −2σ_{3} + 5σ_{2} + 8
6_{59} P_{120} × Z_{17}
0 1  1  −σ_{2} + 1 1 −σ_{3} + 2σ_{2} + 5
1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  −σ_{2} + 1 1 −σ_{3} + 2σ_{2} + 5
6_{60} P_{120} × Z_{23}
0 1  1  −σ_{2} + 1 1  2σ_{2} + 5
1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  −σ_{2} + 1 1  2σ_{2} + 5

c_{i} M ν 3  4  5  6  7
0 1  1  2σ_{2} + 5  3  2σ_{2} + 4
6_{61} P′_{216} 1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  2σ_{2} + 5  3  2σ_{2} + 4
6_{62} M(S^{2}, (3, 2), (3, 1), (3, −2))
0 1  2  2σ_{2} + 5  12  −2σ_{3} + 6σ_{2} + 9
1 -1  −σ_{1}  −2σ_{2} − 5  −6σ_{1}  2σ_{3} − 6σ_{2} − 9
2 0  0  0  0  0
Σ 0 −σ_{1} + 2  0  −6σ_{1} + 12  0
6_{63} M(S^{2}, (3, 2), (3, 2), (3, −2))
0 1  1  σ_{2} + 4  9  −2σ_{3} + 3σ_{2} + 6
1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  σ_{2} + 4  9  −2σ_{3} + 3σ_{2} + 6
6_{64} M(S^{2}, (3, 2), (3, 2), (3, −1))
0 1  1  2σ_{2} + 5  9  σ_{3} + 3
1 0  0  0  0  0
2 0  0  0  0  0
Σ 1  1  2σ_{2} + 5  9  σ_{3} + 3

6_{65} T × I/ ( \begin{matrix}1 − 1 \\ 1 0 \end{matrix} )
0 1  2  σ_{2} + 3  12  0
1 0  0  0  0  0
2 1  -2  σ_{2} + 3  0  0
Σ 2  0  2σ_{2} + 6  12  0

c_{i} M ν 3 4  5  6  7

6_{66} T × I/ ( \begin{matrix}0 1 \\ −1 0 \end{matrix} )
0 1 4  0  12 −σ_{3} + 3σ_{2} + 6
1 -2 0  0  0  2σ_{3} − 6σ_{2} − 12
2 1 0  0  0  −σ_{3} + 3σ_{2} + 6
Σ 0 4  0  12  0

6_{67} T × I/ ( \begin{matrix}0 1 \\ −1 − 1 \end{matrix} )
0 1 2  σ_{2} + 3 12  0
1 0 0  0  0  0
2 1 -2 σ_{2} + 3  0  0
Σ 2 0 2σ_{2} + 6 12  0

6_{68} T × I/ ( \begin{matrix}−1 0 \\ −1 − 1 \end{matrix} )
0 1 4 −σ_{2} + 2 18 −2σ_{3} + 6σ_{2} + 12
1 0 0 0 0 0
2 3 0 −3σ_{2} + 6 18 −6σ_{3} + 18σ_{2} + 36
Σ 4 4 −4σ_{2} + 8 36 −8σ_{3} + 24σ_{2} + 48

6_{69} T × I/ ( \begin{matrix}1 0 \\ 1 1 \end{matrix} )
0 1 4 −σ_{2} + 2 18 −2σ_{3} + 6σ_{2} + 12
1 0 0 0 0 0
2 3 0 −3σ_{2} + 6 18 −6σ_{3} + 18σ_{2} + 36
Σ 4 4 −4σ_{2} + 8 36 −8σ_{3} + 24σ_{2} + 48

c_{i} M ν 3 4  5  6  7

6_{70} T × I/ ( \begin{matrix}−1 0 \\ 0 − 1 \end{matrix} )
0 1 8 4σ_{2} + 12 48  −9σ_{3} + 27σ_{2} + 54
1 0 0 0 0 0
2 7 28 28σ_{2} + 84 252 −63σ_{3} + 189σ_{2} + 378
Σ 8 36 32σ_{2} + 96 300 −72σ_{3} + 216σ_{2} + 432

6_{71} T × I/ ( \begin{matrix}1 0 \\ 0 1 \end{matrix} )
0 1 8 4σ_{2} + 12 48  −9σ_{3} + 27σ_{2} + 54
1 0 0 0 0 0
2 7 28 28σ_{2} + 84 252 −63σ_{3} + 189σ_{2} + 378
Σ 8 36 32σ_{2} + 96 300 −72σ_{3} + 216σ_{2} + 432

6_{72} K ˜×I ∪ K ˜×I/ ( \begin{matrix}−1 0 \\ −1 1 \end{matrix} )
0 1 4 4σ_{2} + 8 28  −9σ_{3} + 19σ_{2} + 30
1 0 0 0 0 0
2 3 12 12σ_{2} + 24 72 −27σ_{3} + 57σ_{2} + 90
Σ 4 16 16σ_{2} + 32 100 −36σ_{3} + 76σ_{2} + 120

c_{i} M ν 3 4  5  6  7

6_{73} K ˜×I ∪ K ˜×I/ ( \begin{matrix}0 1 \\ 1 0 \end{matrix} )
0 1 8 3σ_{2} + 10 46  −6σ_{3} + 22σ_{2} + 44
1 0 0 0 0 0
2 7 24 21σ_{2} + 70 198 −42σ_{3} + 154σ_{2} + 308
Σ 8 32 24σ_{2} + 80 244 −48σ_{3} + 176σ_{2} + 352

6_{74} K ˜×I ∪ K ˜×I/ ( \begin{matrix}1 1 \\ 1 0 \end{matrix} )
0 1 4  σ_{2} + 4  22  −4σ_{3} + 10σ_{2} + 16
1 0 0  0  0  0
2 3 4 3σ_{2} + 12 30 −12σ_{3} + 30σ_{2} + 48
Σ 4 8 4σ_{2} + 16 52 −16σ_{3} + 40σ_{2} + 64

References

[1] Casler, B. G. An imbedding theorem for connected 3-manifolds with boundary. Proc. Amer. Math. Soc. 16 (1965), 559–566.

[2] Matveev, S. V. Transformations of special spines, and the Zeeman conjecture. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 5, 1104– 1116, 1119; translation in Math. USSR-Izv. 31 (1988), no. 2, 423–434.

[3] Matveev, S. V. Complexity theory of three-dimensional manifolds. Acta Appl. Math. 19 (1990), no. 2, 101–130.

[4] Matveev, S. V. Tables of 3-manifolds up to complexity 6. Max-Planck-Institut für Mathematik, preprint MPI 1998-67 (1998).

[5] Milnor, John Groups which act on S^{n} without fixed points. Amer. J. Math. 79 (1957), 623–630.

[6] Ovchinnikov, M. A. The table of 3-manifolds of complexity 7. Preprint, Chelyabinsk State University, 1997.

[7] Rolfsen, Dale Knots and links. Mathematics Lecture Series, No. 7. Publish or Perish, Inc., Berkeley, Calif., 1976. ix+439 pp.

Chelyabinsk State University e-mail: matveev@csu.ru