<Chapter><Heading>Cubical complexes &amp;  permutahedral complexes</Heading>
<Section><Heading>Cubical complexes</Heading>
A <E>finite simplicial complex</E> can be defined to be a CW-subcomplex of the 
canonical regular CW-structure on a simplex <M>\Delta^n</M> of some dimension <M>n</M>. Analogously,  a <E>finite cubical complex</E> is a CW-subcomplex of the regular CW-structure on a cube <M>[0,1]^n</M> of some dimension <M>n</M>. Equivalently, but more conveniently, we can replace the unit interval <M>[0,1]</M> 
by
an interval <M>[0,k]</M> with CW-structure involving <M>2k+1</M> cells, namely one <M>0</M>-cell for each integer <M>0\le j\le k</M> and one <M>1</M>-cell for each open interval <M>(j,j+1)</M> for <M>0\le j\le k-1</M>. A
<E>finite cuical complex</E> <M>M</M> is a CW-subcompex  
<M>M\subset [0,k_1]\times [0,k_2]\times \cdots [0,k_n]</M> of a direct product of intervals, the direct product having the usual direct product CW-structure.
The equivalence of these two definitions follows from the Gray code embedding of a mesh into a hypercube. We say that the cubical complex has <E>ambient dimension</E> <M>n</M>.
A cubical complex <M>M</M> of ambient dimension <M>n</M> is said to be <E>pure</E> if each cell lies in the boundary of an <M>n</M>-cell. In other words, <M>M</M> is pure if it is a union of unit <M>n</M>-cubes in <M>\mathbb R^n</M>, each unit cube having vertices with integer coordinates.


<P/><B>HAP</B> has a datatype for finite cubical complexes, and a slightly different datatype for pure cubical complexes. 

<P/> The following example constructs the granny knot (the sum of a trefoil knot with its reflection) as a <M>3</M>-dimensional pure cubical complex, and then displays it.

<Example>
<#Include SYSTEM "tutex/2.1.txt">
</Example>
<Alt Only="HTML">&lt;img src="images/granny.jpg" align="center" width="350" alt="granny knot"/>
</Alt>

<P/>
Next we construct the complement <M>Y=D^3\setminus \mathring{M}</M> of the interior of the pure cubical complex <M>M</M>. Here <M>D^3</M> is a rectangular region
with   <M>M \subset \mathring{D^3}</M>.
 This pure cubical complex <M>Y</M> is a union of <M>5891</M> unit <M>3</M>-cubes. We contract <M>Y</M> to get a homotopy equivalent pure cubical complex <M>YY</M> consisting of the union of just <M>775</M> unit <M>3</M>-cubes.
Then we convert <M>YY</M> to a regular CW-complex <M>W</M> 
involving <M>11939</M> cells.
We contract <M>W</M> to obtain a homotopy equivalent regular CW-complex
<M>WW</M> involving <M>5993</M> cells.
 Finally we compute the fundamental group of the complement of the granny knot, and use the presentation of this group to establish that the Alexander polynomial <M>P(x)</M> of the granny is
<P/><M>P(x) = x^4-2x^3+3x^2-2x+1 \ .</M>

<Example>
<#Include SYSTEM "tutex/2.2.txt">
</Example>

</Section>

<Section><Heading>Permutahedral complexes</Heading>
<P/>A finite pure cubical complex is a union of finitely many  cubes in 
  a tessellation of  <M>\mathbb R^n</M> by unit cubes. One can also  
tessellate <M>\mathbb R^n</M>  by permutahedra, and we define a 
finite <M>n</M>-dimensional pure <E>permutahedral complex</E> to be a union of 
finitely many permutahdra from such a tessellation. There are two features of pure permutahedral complexes that are particularly useful in some situations:
<List>
<Item> Pure permutahedral complexes are  topological manifolds with boundary.
</Item>
<Item> The method used for finding a smaller pure cubical complex <M>M'</M> 
homotopy equivalent to a given  pure cubical complex <M>M</M> retains the homeomorphism type, and not just the homotopy type, of the space <M>M</M>.</Item>
</List>

<P/><B>Example 1</B>
<P/>To illustrate these  features the following example begins by reading 
in a protein backbone from the online <URL><Link>https://www.rcsb.org/</Link><LinkText>Protein Database</LinkText></URL>, and storing it as a pure cubical complex <M>K</M>. The ends of the protein have been joined, and the homology
 <M>H_i(K,\mathbb Z)=\mathbb Z</M>, <M>i=0,1</M>
 is seen to be that of  a circle. We can thus regard the protein as a knot  
<M>K\subset \mathbb R^3</M>. The protein is visualized as a pure permutahedral complex.

<Example>
<#Include SYSTEM "tutex/2.3a.txt">
</Example>
<Alt Only="HTML">&lt;img src="images/1V2X.jpg" align="center" width="350" alt="knotted protein"/></Alt>

<P/>An alternative method for seeing that the pure permutahedral complex <M>K</M> has the homotopy type of a circle is to note that it is covered by  open permutahedra (small open neighbourhoods of the closed <M>3</M>-dimensional permutahedral titles) and to form the nerve
<M>N=Nerve({\mathcal U})</M> of this open covering <M>\mathcal U</M>. The
 nerve <M>N</M> has the same homotopy type as <M>K</M>. The following commands establish that <M>N</M> is a <M>1</M>-dimensional simplicial complex and display <M>N</M>  as a circular graph.

<Example>
<#Include SYSTEM "tutex/2.3b.txt">
</Example>
<Alt Only="HTML">&lt;img src="images/graphprotein.gif" align="center" width="200" alt="knotted protein"/></Alt>



<P/> The boundary of the pure permutahedral complex <M>K</M>
is a <M>2</M>-dimensional CW-complex <M>B</M> homeomorphic to a torus. We next use the advantageous features of pure permutahedral complexes to compute the homomorphism
<P/><M>\phi\colon \pi_1(B) \rightarrow \pi_1(\mathbb R^3\setminus \mathring{K}),
a \mapsto yx^{-3}y^2x^{-2}yxy^{-1}, b\mapsto yx^{-1}y^{-1}x^2y^{-1}</M>

<P/>where<Br/> 
<M>\pi_1(B)=&lt; a,b\, :\, aba^{-1}b^{-1}=1></M>,<Br/> 
<M>\pi_1(\mathbb R^3\setminus \mathring{K}) \cong &lt; x,y\, :\,  y^2x^{-2}yxy^{-1}=1,  yx^{-2}y^{-1}x(xy^{-1})^2=1></M>.

<Example>
<#Include SYSTEM "tutex/2.3.txt">
</Example>


<P/><B>Example 2</B>
<P/>The next example of  commands begins by readng two synthetic knots from a CSV file (containing the coordinates of the two sequences of vertices) and producing a pure permutahedral complex model of the two knots. The linking number of two knots is given by the low-dimension cup product of the complement of the knots. This linking number is computed to be <M>6</M>.
<P/>
<Alt Only="HTML">&lt;img src="images/twoknots.png" align="center" width="600" alt="two knots"/></Alt>
<Example>
<#Include SYSTEM "tutex/4.9.txt">
</Example>


</Section>

<Section><Heading>Constructing pure cubical and permutahedral complexes</Heading>

<P/> An <M>n</M>-dimensional pure cubical or permutahedral complex can be created from an <M>n</M>-dimensional array of 0s and 1s. The following example creates and displays two <M>3</M>-dimensional complexes.

<Example>
<#Include SYSTEM "tutex/2.4.txt">
</Example>
<Alt Only="HTML">&lt;img src="images/cubicalring.gif" align="center" width="150" alt="cubical ring"/>
</Alt><Alt Only="HTML">&lt;img src="images/permutahedralring.gif" align="center" width="400" alt="permutahedral ring"/>
</Alt>


</Section>

<Section><Heading>Computations in dynamical systems</Heading>


<P/>Pure cubical complexes can be useful for rigourous interval arithmetic calculations in numerical analysis. They can also be useful for trying to estimate 
approximations of certain numerical quantities. To illustrate the latter we consider the  <E>Henon map</E>
<P/><M>f\colon \mathbb R^2 \rightarrow \mathbb R^2,
\left( \begin{array}{cc}
x\\
y
\end{array}\right)
\mapsto
\left( \begin{array}{cc}
y+1-ax^2\\
bx
\\
\end{array}\right) .</M><Br/>

<P/>Starting with <M>(x_0,y_0)=(0,0)</M> and iterating <M>(x_{n+1},y_{n+1}) = f(x_n,y_n)</M> with the parameter values <M>a=1.4</M>, <M>b=0.3</M> one obtains a sequence of points which is known to be
dense in the so called <E>strange attractor</E> <M>{\cal A}</M> of the Henon map. The  first <M>10</M> million  points in this sequence are
 plotted  in the following example, with arithmetic performed to 100 decimal places of accuracy. The sequence is stored as a <M>2</M>-dimensional pure cubical complex where  each <M>2</M>-cell is square of side equal to <M>\epsilon =1/500</M>.

<Example>
<#Include SYSTEM "tutex/2.5.txt">
</Example>
<Alt Only="HTML">&lt;img src="images/henon.png" align="center" width="700" alt="Henon attractor"/> </Alt>

<P/>Repeating the computation but with squares of side <M>\epsilon =1/1000</M>
<Example>
<#Include SYSTEM "tutex/2.6.txt">
</Example>

<P/>
we obtain the heuristic estimate
<P/><M>\delta \simeq  \frac{ \log{ 24949}- \log{ 10287}} {\log{2}} = 1.277 </M>
<P/> for the box-counting dimension of the attractor <M>\cal A</M>.
</Section>
</Chapter>