# Triangulation¶

The main snappy class, namely Manifold, is derived from more basic class below.

class snappy.Triangulation

A Triangulation object represents a compact 3-manifold with torus boundary components, given as an ideal triangulation of the manifold’s interior. A Dehn-filling can be specified for each boundary component, allowing the description of closed 3-manifolds and some orbifolds. For non-orientable 3-manifolds, the boundary components can also be Klein bottles. Two Triangulations are equal (‘==’) if they represent combinatorially isomorphic triangulations. A Triangulation does not have any geometric structure, and usually one works with the subclass Manifold which adds this. Here’s a quick example:

```>>> M = Triangulation('9_42')
>>> M.num_tetrahedra()
5
>>> M.is_orientable()
True
```

A Triangulation can be specified in a number of ways, e.g.

• Triangulation(‘9_42’) : The complement of the knot 9_42 in S^3.

• Triangulation(‘m125(1,2)(4,5)’) : The SnapPea census manifold m125 where the first cusp has Dehn filling (1,2) and the second cusp has filling (4,5).

• Triangulation() : Opens a link editor window where can you specify a link complement.

In general, the specification can be from among the below, with information on Dehn fillings added.

• SnapPea cusped census manifolds: e.g. ‘m123’, ‘s123’, ‘v123’.

• Rolfsen’s table: e.g. ‘4_1’, ‘04_1’, ‘5^2_6’, ‘6_4^7’, ‘L20935’, ‘l104001’.

• Knots and links up to 14 crossings from tabulations by Hoste and Thistlethwaite: e.g. ‘K12a456’ or ‘L13n579’.

• Hoste-Thistlethwaite Knotscape table: e.g. ‘11a17’ or ‘12n345’

• Dowker-Thistlethwaite code: e.g. ‘DT:[(6,8,2,4)]’, ‘DT:dadbcda’

• Once-punctured torus bundles: e.g. ‘b++LLR’, ‘b+-llR’, ‘bo-RRL’, ‘bn+LRLR’

• Fibered manifold associated to a braid: ‘Braid:[1,2,-3,4]’

Here, the braid is thought of as a mapping class of the punctured disc, and this manifold is the corresponding mapping torus. If you want the braid closure, do (1,0) filling of the last cusp.

• From mapping class group data using Twister:

‘Bundle(S_{1,1}, [a0, B1])’ or ‘Splitting(S_{1,0}, [b1, A0], [a0,B1])’

See the help for the ‘twister’ module for more.

• A SnapPea triangulation or link projection file: ‘filename’

The file will be loaded if found in the current directory or the path given by the shell variable SNAPPEA_MANIFOLD_DIRECTORY.

• A Regina-style isomorphism signature, such as ‘dLQbcccdxwb’.

• A string containing the contents of a SnapPea triangulation or link projection file.

alexander_polynomial(**kwargs)

Computes the multivariable Alexander polynomial of the manifold:

```sage: M = Manifold('K12n123')
sage: M.alexander_polynomial()
2*a^6 - 14*a^5 + 34*a^4 - 45*a^3 + 34*a^2 - 14*a + 2

sage: N = Triangulation('v1539(5,1)')
sage: N.alexander_polynomial()
a^2*b + a*b^2 + a*b + a + b
```

Any provided keyword arguments are passed to fundamental_group and so affect the group presentation used in the computation.

homological_longitude(cusp=None)

Returns the peripheral curve in the given cusp, if any, which is homologically trivial (with rational coefficients) in the manifold:

```sage: M = Manifold('m015')
sage: M.homological_longitude()
(2, -1)
```

If no cusp is specified, the default is the first unfilled cusp; if all cusps are filled, the default is the first cusp:

```sage: M = Manifold('L5a1(3,4)(0,0)')
sage: M.homological_longitude()
(0, 1)
```

The components of the next link have nontrivial linking number so there is no such curve:

```sage: W = Manifold('L7a2')
sage: W.homological_longitude(cusp=1) is None
True
```

If every curve in the given cusp is trivial in the rational homology of the manifold, an exception is raised:

```sage: M = Manifold('4_1(1,0)')
sage: M.homological_longitude()
Traceback (most recent call last):
...
ValueError: Every curve on cusp is homologically trivial
```
normal_boundary_slopes(subset='all', algorithm='FXrays')

For a one-cusped manifold, returns all the nonempty boundary slopes of spun normal surfaces. Provided the triangulation supports a genuine hyperbolic structure, then by Thurston and Walsh any strict boundary slope (the boundary of an essential surface which is not a fiber or semifiber) must be listed here.

```>>> M = Manifold('K3_1')
>>> M.normal_boundary_slopes()
[(16, -1), (20, -1), (37, -2)]
```

If the `subset` flag is set to `'kabaya'`, then it only returns boundary slopes associated to vertex surfaces with a quad in every tetrahedron; by Theorem 1.1. of [DG] these are all strict boundary slopes.

```>>> N = Manifold('m113')
>>> N.normal_boundary_slopes()
[(1, 1), (1, 2), (2, -1), (2, 3), (8, 11)]
>>> N.normal_boundary_slopes('kabaya')
[(8, 11)]
```

If the `subset` flag is set to `'brasile'` then it returns only the boundary slopes that are associated to vertex surfaces giving isolated rays in the space of embedded normal surfaces.

```>>> N.normal_boundary_slopes('brasile')
[(1, 2), (8, 11)]
```
normal_surfaces(algorithm='FXrays')

All the vertex spun-normal surfaces in the current triangulation.

```>>> M = Manifold('m004')
>>> M.normal_surfaces()
[<Surface 0: [0, 0] [1, 2] (4, 1)>,
<Surface 1: [0, 1] [1, 2] (4, -1)>,
<Surface 2: [1, 2] [2, 1] (-4, -1)>,
<Surface 3: [2, 2] [2, 1] (-4, 1)>]
```
slice_obstruction_HKL(primes_spec, verbose=False, check_in_S3=True)

For the exterior of a knot in S^3, searches for a topological slicing obstruction from:

Herald, Kirk, Livingston, Math Zeit., 2010 https://dx.doi.org/10.1007/s00209-009-0548-1 https://arxiv.org/abs/0804.1355

The test looks at the cyclic branched covers of the knot of prime order p and the F_q homology thereof where q is an odd prime. The range of such (p, q) pairs searched is given by primes_spec as a list of (p_max, [q_min, q_max]). It returns the pair (p, q) of the first nonzero obstruction found (in which case K is not slice), and otherwise returns None:

```sage: M = Manifold('K12n813')
sage: spec = [(10, [0, 20]), (20, [0, 10])]
sage: M.slice_obstruction_HKL(spec, verbose=True)
Looking at (2, 3) ...
Looking at (3, 2) ...
Looking at (3, 7) ...
(3, 7)
```

If primes_spec is just a pair (p, q) then only that obstruction is checked:

```sage: M.slice_obstruction_HKL((2, 3))
sage: M.slice_obstruction_HKL((3, 7))
(3, 7)
```

Technical note: As implemented, can only get an obstruction when the decomposition of H_1(cover; F_q) into irreducible Z/pZ-modules has no repeat factors. The method of [HKL] can be used more broadly, but other cases requires computing many more twisted Alexander polynomials.