Let \(T_n\) be the \(n \times n\)-Torus. Hence topologically it is torus with \(n^2\) punctures. A cyclic \(n\)-torus cover of degree \(d\) is a normal covering \(X \to T_n\) whose Deck-group is cyclic with \(d\) elements. We can obtain an origami from each cyclic \(n\)-torus cover by appending the map \(p_n: T_n \to T_1\), which sends each square to the singular square of the trivial torus. We call such origamis cyclic torus cover origamis. The functions described in this chapter were programmed and used in the context of [Rog21] and provide a toolkit to work with these special class of origamis.
A cyclic \(n\)-torus cover of degree \(d\) is determined by its monodromy map \(m: \pi_1(T_n) \to \mathbb{Z}/d\mathbb{Z}\). Recall that the fundamental group \(\pi_1(T_n)\) is a free group in \(N = n^2 + 1\) generators. If we choose a basis of \(\pi_1(T_n)\), every cyclic torus cover origami can be described as vector in \((\mathbb{Z}/d\mathbb{Z})^N\). We call this vector monodromy vector with respect to this basis. There are two bases (which we will call \(L\) and \(S\)) of the fundamental group of \(T_n\) which we will use. The base point is chosen as the midpoint of the lower left square.
The basis \(L\) consists of the full horizontal path to the right, the full vertical path upwards and loops around each of the \(n^2 - 1\) corner points of the squares (numbered left to right and bottom up; the loop on the most upper right corner excluded) in this order.
The basis \(S\) consists of picking a maximal set of slits between squares such that \(T_n\) without these is still simply connected. The crossing of one such slit bottom up (for horizontal slits) and left to right (for vertical slits) is then an element of the basis.
Convertion between these two bases is performed with BaseChangeLToS (6.3-1)
See [Rog21] for more information regarding the construction of cyclic torus covers and the bases \(L\) and \(S\).
‣ GeneralizedCyclicTorusCover( n, d, vslits, hslits ) | ( function ) |
Returns: a cyclic torus cover origami
A cyclic torus cover consists of \(d\) copies of the trivial origami \(T_n\). Each of the \(n^2\) fields of \(T_n\) gets assigned a label from \(1\) to \(n^2\) row-wise from left to right and bottom up. Let \(f\) be a field with the label \(k\) in the \(i\)-th copy of the cyclic torus cover. Then \(f\)'s right neighbour's label is determined by determining the usual right neighbour in \(T_n\) and its copy is \(((i+\texttt{vslits[k]}) \bmod d)\). It works in the same way for the upper neighbour and \(\texttt{hslits}\).
gap> GeneralizedCyclicTorusCover(2, 2, [1,0,0,0], [0,0,0,0]); Origami((1,6,5,2)(3,4)(7,8), (1,3)(2,4)(5,7)(6,8), 8)
‣ CombOrigami( n, x, y ) | ( function ) |
Returns: a comb origami, which is a cyclic torus cover of degree 2 specified by a single point \(P=(x,y)\)
A comb origami is a special cyclic torus cover of degree 2, specified by a single point \(P\) on \(T_n\). The coordinates are given in the range \({0,..,n-1}^2\), where the point \((0,0)\) is located in the lower left corner. \(P\) must not be a 2-torsion point, that is, it must not be \((0,0)\), \((n/2, n/2)\), \((n/2,0)\) or \((0,n/2)\). The coordinates are considered modulo \(n\). See [HS07] for more details.
gap> CombOrigami(3, 1, 0); Origami((1,2,3)(4,5,6,13,14,15)(7,8,9)(10,11,12)(16,17,18), (1,4,7)(2,5,8,11,14,17)(3,6,9)(10,13,16)(12,15,18), 18)
‣ SearchForCyclicTorusOrigamiWithVeechGroup( n, p, H ) | ( function ) |
Returns: a monodromy vector in \((\mathbb{Z}/p\mathbb{Z})^{n^2+1}\) representing a cyclic torus cover origami with respect to the basis \(L\) that has \(H\) as its Veech group or \(\texttt{false}\) if no such vector is found.
\(p\) must be prime. \(H\) must be a congruence subgroup of level \(p\) and \(n\) must be \(\geq 2\).
gap> S := [ [ 0, -1 ], [ 1, 0 ] ];; T := [ [ 1, 1 ], [ 0, 1 ] ];; gap> H := ModularSubgroup([S^-2, T*S^-1, T^-1*S^-1]);; gap> SearchForCyclicTorusOrigamiWithVeechGroup(4, 3, H); [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1 ]
‣ CyclicTorusCoverOrigamiS( n, d, v ) | ( function ) |
‣ CyclicTorusCoverOrigamiL( n, d, v ) | ( function ) |
Returns: a cyclic torus cover origami whose monodromy vector with respect to the basis \(S\) (respectively \(L\)) is \(v\).
\(n\) must be \(\geq 2\), \(d\geq 1\) and \(v \in (\mathbb{Z}/d\mathbb{Z})^{n^2+1}\) a vector such that its elements generate \(\mathbb{Z}/d\mathbb{Z}\).
gap> CyclicTorusCoverOrigamiS(2,2,[1,0,1,0,0]); Origami((1,2,5,6)(3,4)(7,8), (1,3,5,7)(2,4)(6,8), 8)
Given a homeomorphism \(f: T_n \to T_n\), one can consider the induced linear map on the homology of \(T_n\). If we furthermore choose two bases of the homology, we can consider this linear map as a matrix. This section computes some of these matrices.
‣ BaseChangeLToS( n ) | ( function ) |
Returns: a matrix \(M = D_{SL}\) representing a change of basis between the bases \(L\) and \(S\).
Returns a matrix corresponding to the change of basis from \(L\) to \(S\) on the homology of \(T_n\). The matrix has the following property: given any cyclic torus cover origami as a monodromy vector \(v\) with respect to the basis \(S\), you may obtain the corresponding monodromy vector with respect to basis \(L\) using \(v \cdot D_{SL}\).
gap> Display(BaseChangeLToS(2)); [ [ 0, 1, 1, -1, 0 ], [ 0, 0, -1, 1, 0 ], [ 1, 0, -1, 0, 1 ], [ 0, 0, 1, 0, -1 ], [ 0, 1, 0, 0, 1 ] ]
‣ TranslationGroupOnHomologyOfTn( n ) | ( function ) |
Returns: the group of translations (as matrices) acting on monodromy vectors with respect to \(L\).
This group of order \(n^2\) is generated by the matrices representing the translation by one square to the right and the translation by one square up with respect to the basis \(L\).
gap> Order(TranslationGroupOnHomologyOfTn(3)); 9
‣ ActionOfTOnHomologyOfTn( n ) | ( function ) |
Returns: a matrix representing the action of the generator \(T\) of \(\mathrm{Sl}_2(\mathbb{Z})\) on the homology of \(T_n\) with respect to \(L\).
The generator \(T\) (shearing to the right) of \(\mathrm{Sl}_2(\mathbb{Z})\) can be viewed as an affine map on \(T_n\) if we assume that the lower left corner is fixed. This function returns the corresponding map on the homology as a matrix with respect to \(L\).
gap> Display(ActionOfTOnHomologyOfTn(2)); [ [ 1, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, -1 ], [ 0, 0, 0, 1, -1 ], [ 0, -1, 0, 0, -1 ] ]
‣ ActionOfSOnHomologyOfTn( n ) | ( function ) |
Returns: a matrix representing the action of the generator \(S\) of \(\mathrm{Sl}_2(\mathbb{Z})\) on the homology of \(T_n\) with respect to \(L\).
The generator \(S\) (rotation by \(\pi/2\) counterclockwise) of \(\mathrm{Sl}_2(\mathbb{Z})\) can be viewed as an affine map on \(T_n\) if we assume that the lower left corner is fixed. This function returns the corresponding map on the homology as a matrix with respect to \(L\).
gap> Display(ActionOfSOnHomologyOfTn(2)); [ [ 0, -1, 0, 0, 0 ], [ 1, 0, 0, 0, 0 ], [ -1, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1 ], [ -1, 0, 0, 1, 0 ] ]
‣ ActionOfMatrixOnHomologyOfTn( n, A ) | ( function ) |
Returns: a matrix representing the action of \(A \in \mathrm{Sl}_2(\mathbb{Z})\) on the homology of \(T_n\) with respect to \(L\).
Any matrix \(A\in\mathrm{Sl}_2(\mathbb{Z})\) can be viewed as an affine map on \(T_n\) if we assume that the lower left corner is fixed. \(A\) must be in \(\mathrm{Sl}_2(\mathbb{Z})\). It is written as a word in the generators \(S\) and \(T\) of \(\mathrm{Sl}_2(\mathbb{Z})\), then the corresponding word with the matrices calculated by ActionOfTOnHomologyOfTn (6.3-3) and ActionOfSOnHomologyOfTn (6.3-4) is taken and returned.
gap> M := [ [ 0, -1 ], [ 1, 1 ] ];; # = S * T gap> Display(ActionOfMatrixOnHomologyOfTn(2, M)); [ [ 0, -1, 0, 0, 0 ], [ 1, 1, 0, 0, 0 ], [ -1, -1, 1, 0, -1 ], [ 0, -1, 0, 0, -1 ], [ -1, -1, 0, 1, -1 ] ]
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