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1. Metrics on Vectors
The data file contains a list L of 400 vectorsi in R3. The following commands produce the 400×400 symmetric matrix whose (i,j)-entry is the Manhattan distance between L[i] and L[j]. (Alternative choices of metric include the Euclidean squared metric and Hamming metric.)
gap> ReadPackage("hap","www/SideLinks/About/data.txt"); #This reads in the list L gap> S:=VectorsToSymmetricMatrix(L,ManhattanMetric);;
The following command uses GraphViz software to display the graph G(S,t) on 400 vertices with edge between i and j precisely when S[i][j] <= tM / 100 where M is the maximum value of the entries in S. Thus the threshold t should be chosen in the range from 0 to 100. We choose t=8. We also choose to give the first 200 vertices a common colour distinct from the remaining vertices. The display shows that the first 200 vertices lie in one path-component of G(S,8), and the remaining 200 vertices lie in a second path-component. Each path-component "has the shape" of a cylinder or annulus.
gap> SymmetricMatDisplay(S,8, [ [1..200], [201..400] ] );
The following commands construct the graph G=G(S,10) and then display it.
gap> M:=Maximum(Maximum(S));; gap> G:=SymmetricMatrixToGraph(S,10*M/100); Graph on 400 vertices. gap> GraphDisplay(G);
We use the term simplicial nerve of G to mean the simplicial complex NG which has the same vertices and edges as G; a collection of vertices is a simplex in NG if and only if each pair of edges in the collection is connected by an edge in G. The following commands determine a subgraph H in G such that the simplicial nerves NG and NH are homotopy equivalent. The commands replace G by H and then display the subgraph.
gap> ContractGraph(G);; gap> G; Graph on 248 vertices. gap> GraphDisplay(G);
The following commands illustrate two methods for calculating the low-dimensional homology of NG. The second method is more efficient in degrees 0 and 1 but has yet to be properly implemented in higher degrees.
gap> #Method One gap> NG:=SimplicialNerveOfGraph(G,3);; gap> NG:=SimplicialComplexToRegularCWComplex(NG); Regular CW-complex of dimension 3 gap> Homology(NG,0); [ 0, 0 ] gap> Homology(NG,1); [ 0, 0 ] gap> #Method Two gap> C:=RipsChainComplex(G,1); Sparse chain complex of length 2 in characteristic 0 . gap> Bettinumbers(C,0); 2 gap> Bettinumbers(C,1); 2
2. Metrics on Permutations
There are a number of standard metrics d(x,y) on permutations x, y such as the Kendall metric (=number of neighbouring transpositions (i,i+1) needed to express x*y^-1), the Cayley metric (= number of transpositions (i,j) needed to express x*y^-1) and the Hamming metric (= #{ i : x*y^-1(i) differs from i } ). The following commands display the Sylow 2-subgroup of S10 with respect to the Cayley metric.
gap> G:=SymmetricGroup(10);; gap> P:=SylowSubgroup(G,2);; gap> P:=Elements(P);; gap> S:=NullMat(Size(P),Size(P));; gap> for i in [1..Size(P)] do > for j in [1..Size(P)] do > S[i][j]:=CayleyMetric(P[i],P[j],10); > od;od; gap> SymmetricMatDisplay(S,50); gap> SymmetricMatDisplay(S,15);
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