Result of Recognition of a Manifold Represented as a Dehn Filling

Manifold is
Dehn filling Q_2(1,-4)          H_1=0

The answer describes the manifold as a Dehn filling of one of the eleven manifolds of complexity at most 3, with boundary a torus and hyperbolic interior. These manifolds are named by Q_1, Q_2, ..., Q_11. These are complement spaces of knots in 3-manifolds of genus 0 or 1. For instance, Q_2 is the complement of the figure-eight knot. In the picture below you can see the 14 special spines of these manifolds used by the program. The first eleven spines P_1 to P_11 represent the manifolds Q_1 to Q_11 respectively, P_12 is again a spine of manifold Q_6, P_13 is a spine of manifold Q_1, and P_14 is a spine of Q_2.

The program is also able recognize Dehn fillings of hyperbolic manifolds with at most 4 cusps and complexity up to 7. For this purpose we use the list of hyperbolic manifolds of complexity 4 to 7 having one or two cusps. (In fact, a few manifolds of complexity 8-10 are also used, so sometimes the answer can be obtained for hyperbolic manifolds complexity greater than 7.) This list was published by J. Weeks. Please note that our notation for these manifolds is different from the original one.

Hyperbolic manifolds with 1 cusp are denoted as Qk_n, where k is the complexity and n is a number in order. Hyperbolic manifolds with more than one cusp are denoted by Qk^l_n, where K is the complexity, l is the number of cusps, n is the number in order. The order starts from 1 for each number of cusps, for instance, we have Q6_1, Q6^2_1, Q6^3_1.

The total number of manifolds obtained from the table of J. Weeks is:

With 1 cusp:
complexity 2 -- 2,
complexity 3 -- 9,
complexity 4 -- 52,
complexity 5 -- 223,
complexity 6 -- 913,
complexity 7 -- 3388.

With 2 cusps:

complexity 4 -- 4,
complexity 5 -- 11,
complexity 6 -- 48,
complexity 7 -- 162.

With 3 cusps:

complexity 6 -- 1,
complexity 7 -- 2.

If the program found a presentation of a given manifold using some of these known hyperbolic manifolds with two cusps, the answer would be something like this:

The notation "i: Free (a b) (c d)" means that the i-th boundary torus is free and the canonical coordinate system on it and the coordinate system on the boundary of the recognized manifold are related via the matrix (a b) (c d), where (a b) and (c d) are rows. Thus, the manifolds in the first and the second lines have boundaries that consist of a single torus, while the manifold in the third line is closed.

If the result of the recognition is a hyperbolic manifold with two or more cusps and it is a manifold from the table then the answer will have a more complicated form. Namely, the program tries to establish a correspondence between the cusps of the initial manifold and those of atoms taken from the table. For example, let us consider the following manifold:

link
signs 1 1 1 1 1 1 1
code 4 -3 2 -1
framing 1
code 5 -6 3 -4
code 1 -2 6 -5
end

If we first perform (within the dialog) a crude piercing and then close dialog and run recognition, we soon obtained the following presentations:

         +3D^3          H_1=Z+Z
 1
 Edge 1 { Solid torus 3 (1,1) , Exceptional vertex 4 L6^3_1 } (1 0) (0 1)
 Virtual edge2 { Virtual vertex 1  , Exceptional vertex 4 L6^3_1 } (1 0) (0 1)
 Virtual edge3 { Virtual vertex 2  , Exceptional vertex 4 L6^3_1 } (1 0) (0 1)

 2
 Edge 1 { Solid torus 3 (-1,2) , Exceptional vertex 4 Q6^3_1 } (1 0) (0 1)
 Virtual edge2 { Virtual vertex 1  , Exceptional vertex 4 Q6^3_1 } (0 1) (1 2)
 Virtual edge3 { Virtual vertex 2  , Exceptional vertex 4 Q6^3_1 } (-1 -2) (0 1)

 3
 Virtual edge1 { Virtual vertex 1  , Exceptional vertex 4 Q6^3_1 } (1 0) (-1 -1)
 Virtual edge2 { Virtual vertex 2  , Exceptional vertex 4 Q6^3_1 } (0 1) (1 2)
 Edge 3 { Solid torus 3 (-3,1) , Exceptional vertex 4 Q6^3_1 } (1 0) (0 1) 

Here the i-th virtual vertex corresponds to the i-th toral component of the boundary of the manifold (the numbering of boundary components can be viewed via "Information|Boundary"), and the j-th edge corresponds to the j-th cusp.