  
  [1X15 [33X[0;0YRegular CW-structure on knots (written by Kelvin Killeen)[133X[101X
  
  
  [1X15.1 [33X[0;0YKnot complements in the 3-ball[133X[101X
  
  [33X[0;0YWhile  methods for endowing knot complements with CW-structure already exist
  in HAP (see section 2.1), they often result in a large number of cells which
  can make computing with them taxing. The following example shows how one can
  obtain  a comparatively small 3-dimensional regular CW-complex corresponding
  to  the  complement  of  a  thickened trefoil knot from an arc presentation.
  Recall that an arc presentation is encoded in HAP as a list of integer pairs
  corresponding  to  the position of the endpoints of each horizontal arc in a
  grid.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xk_:=PureCubicalKnot(3,1);                  [127X[104X
    [4X[28Xprime knot 1 with 3 crossings[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xarc:=ArcPresentation(k_);                  [127X[104X
    [4X[28X[ [ 2, 5 ], [ 1, 3 ], [ 2, 4 ], [ 3, 5 ], [ 1, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27Xk_:=RegularCWComplex(PureComplexComplement(k_));[127X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(k_);[127X[104X
    [4X[28X13291[128X[104X
    [4X[25Xgap>[125X [27Xk:=KnotComplement(arc);                                         [127X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(k);[127X[104X
    [4X[28X395[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAn optional argument of [10X"rand"[110X in the [10XKnotComplement[110X function randomises the
  order  in  which [22X2[122X-cells are added to the complex. This allows for alternate
  presentations of the knot group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xarc:=ArcPresentation(PureCubicalKnot(3,1));[127X[104X
    [4X[28X[ [ 2, 5 ], [ 1, 3 ], [ 2, 4 ], [ 3, 5 ], [ 1, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27Xk:=KnotComplement(arc,"rand");[127X[104X
    [4X[28XRandom 2-cell selection is enabled.[128X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xg:=FundamentalGroup(k); RelatorsOfFpGroup(g); [127X[104X
    [4X[28X#I  there are 2 generators and 1 relator of total length 6[128X[104X
    [4X[28X<fp group of size infinity on the generators [ f1, f2 ]>[128X[104X
    [4X[28X[ f2^-1*f1*f2^-1*f1^-1*f2*f1^-1 ][128X[104X
    [4X[25Xgap>[125X [27Xk:=KnotComplement(arc,"rand");               [127X[104X
    [4X[28XRandom 2-cell selection is enabled.[128X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xg:=FundamentalGroup(k); RelatorsOfFpGroup(g);[127X[104X
    [4X[28X#I  there are 2 generators and 1 relator of total length 7[128X[104X
    [4X[28X<fp group of size infinity on the generators [ f1, f2 ]>[128X[104X
    [4X[28X[ f1*f2^-2*f1*f2*f1^-1*f2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  is  often  useful  to  obtain an inclusion of regular CW-complexes [22Xι : ∂
  (N(K))  ↪ B^3 backslash N(K)[122X from the boundary of a tubular neighbourhood of
  some  knot  [22XN(K)[122X  into  its complement in the [22X3[122X-ball [22XB^3 backslash N(K)[122X. The
  below example does this for the first prime knot on 11 crossings.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xarc:=ArcPresentation(PureCubicalKnot(11,1));[127X[104X
    [4X[28X[ [ 2, 9 ], [ 1, 3 ], [ 2, 6 ], [ 4, 7 ], [ 3, 5 ], [ 6, 10 ], [ 4, 8 ], [128X[104X
    [4X[28X  [ 9, 11 ], [ 7, 10 ], [ 1, 8 ], [ 5, 11 ] ][128X[104X
    [4X[25Xgap>[125X [27Xk:=KnotComplementWithBoundary(arc);[127X[104X
    [4X[28XMap of regular CW-complexes[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(Source(i));[127X[104X
    [4X[28X616[128X[104X
    [4X[25Xgap>[125X [27XSize(Target(i));[127X[104X
    [4X[28X1043[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  we  can  add  [22Xn[122X-cells  to regular CW-complexes by specifying the
  [22X(n-1)[122X-cells in their boundaries and [22X(n+1)[122X-cells in their coboundaries.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xk:=KnotComplement([[1,2],[1,2]])!.boundaries;;[127X[104X
    [4X[25Xgap>[125X [27XHomology(RegularCWComplex(k),0);[127X[104X
    [4X[28X[ 0 ][128X[104X
    [4X[25Xgap>[125X [27XAddCell(k,0,[0],[]);                          [127X[104X
    [4X[25Xgap>[125X [27XHomology(RegularCWComplex(k),0);[127X[104X
    [4X[28X[ 0, 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X15.2 [33X[0;0YTubular neighbourhoods[133X[101X
  
  [33X[0;0YLet  [22XY[122X  denote a CW-subcomplex of a regular CW-complex [22XX[122X and let [22XN(Y)[122X denote
  an   open  tubular  neighbourhood  of  [22XY[122X.  Given  an  inclusion  of  regular
  CW-complexes  [22Xf  : Y ↪ X[122X, this algorithm describes a procedure for obtaining
  the  associated  inclusion  [22Xf'  :  ∂  C ↪ C[122X where [22XC=X backslash N(Y)[122X and [22X∂ C[122X
  denotes the boundary of [22XC[122X. The following is also assumed:[133X
  
  [33X[0;0YLet  [22Xe^n[122X  denote a cell of [22XX backslash Y[122X of dimension [22Xn[122X with [22Xbare^n[122X denoting
  its  closure.  For  each  [22Xn[122X-cell,  all  of  the  connected components of the
  subcomplex [22Xbare^n ∩ Y[122X are contractible.[133X
  
  [33X[0;0YSome  additional  terminology  and  notation  is  needed  to  describe  this
  algorithm.  The  output  regular CW-complex [22XX backslash N(Y)[122X consists of the
  cell  complex  [22XX  backslash  Y[122X  as well as some additional cells to maintain
  regularity. A cell of [22XX backslash N(Y)[122X is referred to as [13Xinternal[113X if it lies
  in [22XX backslash Y[122X, it is [13Xexternal[113X otherwise. Let [22Xbare^n[122X denote the closure in
  [22XX[122X  of  an internal cell [22Xe^n[122X. Note that [22Xbare^n[122X is a CW-subcomplex of [22XX[122X and so
  is the intersection [22Xbare^n ∩ Y[122X which can be expressed as the union[133X
  
  [33X[0;0Y[22Xbare^n ∩ Y = A_1 ∪ A_2 ∪ ⋯ ∪ A_k[122X[133X
  
  [33X[0;0Yof  its  path components [22XA_i[122X all of which are CW-subcomplexes of [22XY[122X. For each
  [22Xn[122X-cell  of  [22XX  backslash  Y[122X  there is one internal [22Xn[122X-cell [22Xe^n[122X of [22XX backslash
  N(Y)[122X.  For  [22Xn  ≥  1[122X there is also one external [22X(n-1)[122X-cell [22Xf^e^n_A_i[122X for each
  path  component  [22XA_i[122X of [22Xbare^n ∩ Y[122X. Lastly, we need a method for determining
  the homological boundary of the internal and external cells:[133X
  
  [33X[0;0Y[22X∙[122X  The  boundary  of  an  internal [22Xn[122X-cell [22Xe^n[122X consists of all those internal
  [22X(n-1)[122X-cells of [22Xbare^n[122X together with all external [22X(n-1)[122X-cells [22Xf^e^n_A_i[122X where
  [22XA_i[122X is a path component of [22Xbare^n ∩ Y[122X.[133X
  
  [33X[0;0Y[22X∙[122X  The  boundary  of  an external [22X(n-1)[122X-cell [22Xf^e^n_A_i[122X consists of all those
  external [22X(n-2)[122X-cells [22Xf^e^n-1}_B_j[122X where [22Xe^n-1[122X is an [22X(n-1)[122X-cell of [22Xbare^n[122X and
  [22XB_j ⊆ A_i[122X is a path component of [22XA_i[122X.[133X
  
  [33X[0;0YThe following three steps comprise the algorithm.[133X
  
  [33X[0;0Y[22X(1)[122X  For  each  internal  [22Xn[122X-cell [22Xe^n ⊂ X backslash Y[122X, compute the CW-complex
  [22Xbare^n  ∩  Y[122X  as  a  union  of  path  components  [22XA_1  ∪ A_2 ∪ ⋯ ∪ A_k[122X. This
  information can be used to determine the number of cells of [22XX backslash N(Y)[122X
  in each dimension.[133X
  
  [33X[0;0Y[22X(2)[122X Create a list [22XB=[ [ ], [ ], ..., [ ] ][122X of length [22XdimX +1[122X.[133X
  
  [33X[0;0Y[22X(3)[122X  For  [22X0  ≤ n ≤ dimX[122X set [22XB[n+1]=[ b_1, b_2, ..., b_α_n ][122X where [22Xα_n[122X is the
  number  of  [22Xn[122X-cells  in  [22XX  backslash  N(Y)[122X  and  [22Xb_i[122X  is a list of integers
  describing  the  [22X(n-1)[122X-cells  of  the [22Xi ^ th[122X [22Xn[122X-cell of [22XX backslash N(Y)[122X. The
  internal  cells  will  always  be  listed  before the external cells in each
  sublist. Return B as a regular CW-complex.[133X
  
  [33X[0;0YThe  following example computes the tubular neighbourhood of a [22X1[122X-dimensional
  subcomplex  of  a  [22X3[122X-dimensional  complex  corresponding  to  the  Hopf link
  embedded in the closed [22X3[122X-ball.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xarc:=[[2,4],[1,3],[2,4],[1,3]];;            [127X[104X
    [4X[25Xgap>[125X [27Xf:=ArcPresentationToKnottedOneComplex(arc);[127X[104X
    [4X[28XMap of regular CW-complexes[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xcomp:=RegularCWComplexComplement(f);[127X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X151 out of 151 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that the output of this algorithm is just a regular CW-complex, not an
  inclusion  map.  The  function  [10XBoundaryMap[110X  can  be  employed to obtain the
  boundary  of  a pure complex. This results in three path components for this
  example: two corresponding to the boundary of the knotted tori and the other
  corresponding  to the boundary of the [22X3[122X-ball in which the link was embedded.
  These  path  components  can  be  obtained  as individual CW-subcomplexes if
  desired.  A CW-subcomplex is represented in HAP as a list [22X[X,s][122X where [22XX[122X is a
  regular  CW-complex  and  [22Xs[122X is a list of length [22Xn[122X whose [22Xi^th[122X entry lists the
  indexing   of   each  [22X(i-1)[122X-cell  of  the  [22Xn[122X-dimensional  subcomplex  of  [22XX[122X.
  CW-subcomplexes   and   CW   maps   can  be  converted  between  each  other
  interchangeably.  This  next  example  obtains the inclusion detailed in the
  above  algorithm, finds the path components of the source of said inclusion,
  shows  that  they  are  in  fact  disjoint,  and then obtains the first four
  integral homology groups of each component.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf_:=BoundaryMap(comp);[127X[104X
    [4X[28XMap of regular CW-complexes[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xf_:=RegularCWMapToCWSubcomplex(f_);;[127X[104X
    [4X[25Xgap>[125X [27Xpaths:=PathComponentsCWSubcomplex(f_);[127X[104X
    [4X[28X[ [ Regular CW-complex of dimension 3[128X[104X
    [4X[28X        , [128X[104X
    [4X[28X      [ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 20 ], [128X[104X
    [4X[28X          [ 1, 2, 3, 4, 5, 6, 13, 14, 15, 16, 17, 18, 33, 34, 35, 46, 47, 48 [128X[104X
    [4X[28X             ], [ 11, 12, 13, 14, 15, 16, 35, 36 ] ] ], [128X[104X
    [4X[28X  [ Regular CW-complex of dimension 3[128X[104X
    [4X[28X        , [ [ 21, 24, 25, 27, 30, 31, 32, 37, 38, 39, 40, 43, 45, 46, 48 ], [128X[104X
    [4X[28X          [ 49, 51, 53, 56, 57, 59, 61, 63, 65, 67, 69, 71, 73, 74, 76, 79, [128X[104X
    [4X[28X              82, 83, 86, 87, 90, 91 ], [ 37, 39, 41, 44, 45, 47, 49 ] ] ], [128X[104X
    [4X[28X  [ Regular CW-complex of dimension 3[128X[104X
    [4X[28X        , [ [ 22, 23, 26, 28, 29, 33, 34, 35, 36, 41, 42, 44, 47, 49, 50 ], [128X[104X
    [4X[28X          [ 50, 52, 54, 55, 58, 60, 62, 64, 66, 68, 70, 72, 75, 77, 78, 80, [128X[104X
    [4X[28X              81, 84, 85, 88, 89, 92 ], [ 38, 40, 42, 43, 46, 48, 50 ] ] ] ][128X[104X
    [4X[25Xgap>[125X [27Xpaths:=List(paths,CWSubcomplexToRegularCWMap);[127X[104X
    [4X[28X[ Map of regular CW-complexes[128X[104X
    [4X[28X    , Map of regular CW-complexes[128X[104X
    [4X[28X    , Map of regular CW-complexes[128X[104X
    [4X[28X     ][128X[104X
    [4X[25Xgap>[125X [27XList([1..3],x->List(Difference([1..3],[x]),y->IntersectionCWSubcomplex(paths[x],paths[y])));[127X[104X
    [4X[28X[ [ [ Regular CW-complex of dimension 3[128X[104X
    [4X[28X            , [ [  ], [  ], [  ] ] ], [ Regular CW-complex of dimension 3[128X[104X
    [4X[28X            , [ [  ], [  ], [  ] ] ] ], [ [ Regular CW-complex of dimension 3[128X[104X
    [4X[28X            , [ [  ], [  ], [  ] ] ], [ Regular CW-complex of dimension 3[128X[104X
    [4X[28X            , [ [  ], [  ], [  ] ] ] ], [ [ Regular CW-complex of dimension 3[128X[104X
    [4X[28X            , [ [  ], [  ], [  ] ] ], [ Regular CW-complex of dimension 3[128X[104X
    [4X[28X            , [ [  ], [  ], [  ] ] ] ] ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XList(paths,x->List([0..3],y->Homology(Source(x),y)));[127X[104X
    [4X[28X[ [ [ 0 ], [  ], [ 0 ], [  ] ], [ [ 0 ], [ 0, 0 ], [ 0 ], [  ] ], [128X[104X
    [4X[28X  [ [ 0 ], [ 0, 0 ], [ 0 ], [  ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAs previously mentioned, for the tubular neighbourhood algorithm to work, we
  require  that  no  external  cells yield non-contractible path-components in
  their intersection with the subcomplex. If this is ever the case then we can
  subdivide  the  offending  cell  to  prevent  this  from  happening. We have
  implemented  two  subdivision  algorithms  in  HAP,  one for barycentrically
  subdividing  a  given  cell, and the other for subdividing an [22Xn[122X-cell into as
  many   [22Xn[122X-cells  as  there  are  [22X(n-1)[122X-cells  in  its  boundary.  Barycentric
  subdivision  is  integrated into the [10XRegularCWComplexComplement[110X function and
  will  be  performed  automatically  as required. The following example shows
  this   automatic  subdivision  running  via  the  complement  of  a  tubular
  neighbourhood  of the unknot, then obtains an inclusion map from the closure
  of  an  arbitrary [22X3[122X-cell of this complex and then compares the difference in
  size  of  the two different subdivisions of a 2-cell in the boundary of this
  [22X3[122X-cell.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xarc:=[[1,2],[1,2]];;[127X[104X
    [4X[25Xgap>[125X [27Xunknot:=ArcPresentationToKnottedOneComplex(arc);[127X[104X
    [4X[28XMap of regular CW-complexes[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xf:=RegularCWComplexComplement(unknot);[127X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X79 out of 79 cells tested.[128X[104X
    [4X[28XSubdividing 3 cell(s):[128X[104X
    [4X[28X100% complete. [128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X145 out of 145 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XRegular CW-complex of dimension 3[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xf:=Objectify(HapRegularCWMap,rec(source:=f,target:=f,mapping:={i,j}->j));    [127X[104X
    [4X[28XMap of regular CW-complexes[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xclosure:=ClosureCWCell(Target(f),3,1);[127X[104X
    [4X[28X[ Regular CW-complex of dimension 3[128X[104X
    [4X[28X    , [128X[104X
    [4X[28X  [ [ 1, 2, 3, 4, 7, 8, 9, 10, 11, 13, 14, 20, 21, 22, 23, 25 ], [128X[104X
    [4X[28X      [ 1, 2, 3, 7, 8, 9, 10, 11, 15, 16, 17, 20, 21, 22, 23, 24, 25, 27, 28, 55, 58, 59, [128X[104X
    [4X[28X          60, 63 ], [ 1, 4, 7, 8, 9, 13, 14, 15, 18, 52 ], [ 1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XSize(Target(f));                                          [127X[104X
    [4X[28X195[128X[104X
    [4X[25Xgap>[125X [27XSize(Target(BarycentricallySubdivideCell(f,2,1)));        [127X[104X
    [4X[28X231[128X[104X
    [4X[25Xgap>[125X [27XSize(Target(SubdivideCell(f,2,1)));        [127X[104X
    [4X[28X207[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X15.3 [33X[0;0YKnotted surface complements in the 4-ball[133X[101X
  
  [33X[0;0YA  construction  of Satoh's, the tube map, associates a ribbon torus-knot to
  virtual knot diagrams. A virtual knot diagram differs from a knot diagram in
  that  it  allows for a third type of crossing, a virtual crossing. The image
  of  such  a  crossing  via  the tube map is two tori which pass through each
  other.  An  arc  diagram  is  a triple of lists [10X[arc,cross,cols][110X that encode
  virtual knot diagrams. [10Xarc[110X is an arc presentation. [10Xcross[110X is a list of length
  the number of crossings in the knot associated to the arc presentation whose
  entries  are  [22X-1,0[122X  or  [22X1[122X  corresponding to an undercrossing (horizontal arc
  underneath  vertical  arc),  a  virtual  crossing  (depicted by intersecting
  horizontal  and  vertical  arcs)  and  an overcrossing (horizontal arc above
  vertical arc) respectively. [10Xcols[110X is a list of length the number of [22X0[122X entries
  in  [10Xcross[110X  and  its  entries  are  [22X1,2,3[122X  or  [22X4[122X.  It  describes the types of
  'colourings'  we  assign to the virtual crossings. We interpret each integer
  as the change in 4-dimensional height information as represented by a colour
  scale  from blue (lower down in 4-space), to green (0 level), to red (higher
  up  in  4-space). Without loss of generality, we impose that at each virtual
  crossing,  the  vertical  arc  passes  through  the  horizontal arc. Thus, [22X1[122X
  corresponds  to  the  vertical  bar  entering the horizontal bar as blue and
  leaving  as  blue,  [22X2[122X  corresponds to entering as blue and leaving as red, [22X3[122X
  corresponds  to  entering  as  red  and leaving as blue and [22X4[122X corresponds to
  entering  and leaving as red. A coloured arc diagram can be visualised using
  the [10XViewColouredArcDiagram[110X function.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xarc:=ArcPresentation(PureCubicalKnot(6,1));[127X[104X
    [4X[28X[ [ 5, 8 ], [ 4, 6 ], [ 3, 5 ], [ 2, 4 ], [ 1, 3 ], [ 2, 7 ], [ 6, 8 ], [ 1, 7 ] ][128X[104X
    [4X[25Xgap>[125X [27Xcross:=[0,0,1,-1,-1,0];;[127X[104X
    [4X[25Xgap>[125X [27Xcols:=[1,4,3];;[127X[104X
    [4X[25Xgap>[125X [27XViewColouredArcDiagram(arc,cross,cols);  [127X[104X
    [4X[28Xconvert-im6.q16: pixels are not authentic `/tmp/HAPtmpImage.txt' @ error/cache.c/QueueAuthenticPixelCacheNexus/4381.[128X[104X
    [4X[28X[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTowards obtaining a regular CW-decomposition of ribbon torus-knots, we first
  begin  by  embedding  a  self-intersecting  knotted torus in the 3-ball. The
  function  [10XArcDiagramToTubularSurface[110X  inputs  a  coloured  arc  diagram  and
  outputs    an   inclusion   from   the   boundary   of   some   (potentially
  self-intersecting)   torus   in   the  [22X3[122X-ball.  By  inputting  just  an  arc
  presentation,    one    can   obtain   an   inclusion   identical   to   the
  [10XKnotComplementWithBoundary[110X function. By additionally inputting a list of [22X-1[122Xs
  and  [22X1[122Xs,  one  can obtain an inclusion similar to [10XKnotComplementWithBoundary[110X
  but  where  there  is  extra  freedom  in determining whether or not a given
  crossing  is  an  under/overcrossing.  If  one  inputs both of the above but
  includes  [22X0[122X  entries in the [10Xcross[110X list and includes the list of colours, the
  output  is  then  an inclusion from an embedded self-intersecting torus into
  the   3-ball   where   each   [22X2[122X-cell   (the  top-dimensional  cells  of  the
  self-intersecting surface) is assigned a colour.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtub:=ArcDiagramToTubularSurface(arc);        [127X[104X
    [4X[28XMap of regular CW-complexes[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xtub:=ArcDiagramToTubularSurface([arc,cross]);[127X[104X
    [4X[28XMap of regular CW-complexes[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xtub:=ArcDiagramToTubularSurface([arc,cross,cols]);[127X[104X
    [4X[28XMap of regular CW-complexes[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XList([1..Length(Source(tub)!.boundaries[3])],x->tub!.colour(2,tub!.mapping(2,x)));[127X[104X
    [4X[28X[ [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [128X[104X
    [4X[28X  [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [128X[104X
    [4X[28X  [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [128X[104X
    [4X[28X  [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [128X[104X
    [4X[28X  [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ -1 ], [ -1 ], [128X[104X
    [4X[28X  [ 0 ], [ 0 ], [ -1 ], [ -1 ], [ -1 ], [ -1 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 1 ], [ 1 ], [128X[104X
    [4X[28X  [ 0 ], [ 0 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 1 ], [ 1 ], [ 0 ], [128X[104X
    [4X[28X  [ 0 ], [ -1 ], [ -1 ], [ 1 ], [ -1 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ -1 ], [ -1 ], [ 0 ], [128X[104X
    [4X[28X  [ 1 ], [ 1 ], [ 0 ], [ 0 ], [ 0 ], [ 0 ], [ 1 ], [ -1 ], [ 0 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFrom this self-intersecting surface with colour, we can lift it to a surface
  without  self-intersections  in  [22XR^4[122X.  We  do this by constructing a regular
  CW-complex of the direct product [22XB^3 × [a,b][122X where [22XB^3[122X denotes the [22X3[122X-ball, [22Xa[122X
  is [22X1[122X less than the smallest integer assigned to a cell by the colouring, and
  [22Xb[122X is [22X1[122X greater than the largest integer assigned to a cell by the colouring.
  The  subcomplex  of  the direct product corresponding to the surface without
  intersection  can be obtained using the colouring with additional care taken
  to not lift any 1-cells arising as double-point singularities. The following
  example  constructs  the  complement  of a ribbon torus-link embedded in [22XR^4[122X
  obtained  from  the  Hopf link with one virtual crossing and then calculates
  some invariants of the resulting space. We compare the size of this complex,
  as  well  as how long it takes to obtain the same invariants, with a cubical
  complex of the same space. As barycentric subdivision can massively increase
  the  size  of  the  cell  complex, the below method sequentially obtains the
  tubular  neighbourhood  of  the  entire  subcomplex by obtaining the tubular
  neighbourhood  of each individual [22X2[122X-cell. This has yet to be optimised so it
  currently takes some time to complete.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xarc:=[[2,4],[1,3],[2,4],[1,3]];;                [127X[104X
    [4X[25Xgap>[125X [27Xtub:=ArcDiagramToTubularSurface([arc,[0,-1],[2]]);[127X[104X
    [4X[28XMap of regular CW-complexes[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xtub:=LiftColouredSurface(tub);[127X[104X
    [4X[28XMap of regular CW-complexes[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XDimension(Source(tub));[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XDimension(Source(tub));[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27Xmap:=RegularCWMapToCWSubcomplex(tub);;[127X[104X
    [4X[25Xgap>[125X [27Xsub:=SortedList(map[2][3]);;[127X[104X
    [4X[25Xgap>[125X [27Xsub:=List(sub,x->x-(Position(sub,x)-1));;[127X[104X
    [4X[25Xgap>[125X [27Xclsr:=ClosureCWCell(map[1],2,sub[1])[2];;[127X[104X
    [4X[25Xgap>[125X [27Xseq:=CWSubcomplexToRegularCWMap([map[1],clsr]);;[127X[104X
    [4X[25Xgap>[125X [27Xtub:=RegularCWComplexComplement(seq);[127X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X3501 out of 3501 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[25Xgap>[125X [27Xfor i in [2..Length(sub)] do[127X[104X
    [4X[25X>[125X [27X    clsr:=ClosureCWCell(tub,2,sub[i])[2];;[127X[104X
    [4X[25X>[125X [27X    seq:=CWSubcomplexToRegularCWMap([tub,clsr]);;[127X[104X
    [4X[25X>[125X [27X    tub:=RegularCWComplexComplement(seq);[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X3612 out of 3612 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X3693 out of 3693 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X3871 out of 3871 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X3925 out of 3925 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X4084 out of 4084 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X4216 out of 4216 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X4348 out of 4348 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X4529 out of 4529 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X4688 out of 4688 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X4723 out of 4723 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X4918 out of 4918 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X5107 out of 5107 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X5269 out of 5269 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X5401 out of 5401 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X5548 out of 5548 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X5702 out of 5702 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X5846 out of 5846 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X6027 out of 6027 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X6089 out of 6089 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X6124 out of 6124 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X6159 out of 6159 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X6349 out of 6349 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X6467 out of 6467 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X6639 out of 6639 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X6757 out of 6757 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X6962 out of 6962 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X7052 out of 7052 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X7242 out of 7242 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X7360 out of 7360 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X7470 out of 7470 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X7561 out of 7561 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X7624 out of 7624 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X7764 out of 7764 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X7904 out of 7904 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X7979 out of 7979 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X8024 out of 8024 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X8086 out of 8086 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X8148 out of 8148 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X8202 out of 8202 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X8396 out of 8396 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X8534 out of 8534 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X8625 out of 8625 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X8736 out of 8736 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X8817 out of 8817 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X8983 out of 8983 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X9073 out of 9073 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X9218 out of 9218 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X9323 out of 9323 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X9442 out of 9442 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X9487 out of 9487 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X9538 out of 9538 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X9583 out of 9583 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[28XTesting contractibility...[128X[104X
    [4X[28X9634 out of 9634 cells tested.[128X[104X
    [4X[28XThe input is compatible with this algorithm.[128X[104X
    [4X[25Xgap>[125X [27XSize(tub);      [127X[104X
    [4X[28X9685[128X[104X
    [4X[25Xgap>[125X [27Xtotal_time_1:=0;;[127X[104X
    [4X[25Xgap>[125X [27XList([0..4],x->Homology(tub,x)); total_time_1:=total_time_1+time;;[127X[104X
    [4X[28X[ [ 0 ], [ 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0 ], [  ] ][128X[104X
    [4X[25Xgap>[125X [27Xc:=ChainComplexOfUniversalCover(tub);; total_time_1:=total_time_1+time;;[127X[104X
    [4X[25Xgap>[125X [27Xl:=Filtered(LowIndexSubgroups(c!.group,5),g->Index(c!.group,g)=5);; total_time_1:=total_time_1+time;;[127X[104X
    [4X[25Xgap>[125X [27Xinv:=Set(l,g->Homology(TensorWithIntegersOverSubgroup(c,g),2)); total_time_1:=total_time_1+time;;[127X[104X
    [4X[28X[ [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] [128X[104X
    [4X[28X ][128X[104X
    [4X[25Xgap>[125X [27Xtotal_time_1;[127X[104X
    [4X[28X3407[128X[104X
    [4X[25Xgap>[125X [27Xhopf:=PureComplexComplement(HopfSatohSurface());;[127X[104X
    [4X[25Xgap>[125X [27Xhopf:=RegularCWComplex(hopf);;[127X[104X
    [4X[25Xgap>[125X [27XSize(hopf);[127X[104X
    [4X[28X4508573[128X[104X
    [4X[25Xgap>[125X [27Xtotal_time_2:=0;;[127X[104X
    [4X[25Xgap>[125X [27Xc_:=ChainComplexOfUniversalCover(hopf);; total_time_2:=total_time_2+time;;[127X[104X
    [4X[25Xgap>[125X [27Xl_:=Filtered(LowIndexSubgroups(c_!.group,5),g->Index(c_!.group,g)=5);; total_time_2:=total_time_2+time;;[127X[104X
    [4X[25Xgap>[125X [27Xinv_:=Set(l_,g->Homology(TensorWithIntegersOverSubgroup(c_,g),2));; total_time_2:=total_time_2+time;;[127X[104X
    [4X[25Xgap>[125X [27Xtotal_time_2;[127X[104X
    [4X[28X1116000[128X[104X
    [4X[25Xgap>[125X [27Xinv_=inv;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
