Added: Knots and 2nd quandle homology group of some quandles

homology/homology.gd

@@ -1,2 +1,2 @@
-DeclareGlobalFunction("2ndQuandlehomologyGenerators");
+DeclareGlobalFunction("2ndQuandleHomologyGenerators");
 DeclareGlobalFunction("2ndQuandleCohomologyGenerators");

homology/homology.gi

@@ -105,8 +105,8 @@ InstallGlobalFunction("2ndQuandleHomologyGenerators", function(size, number)
 
   known := IsBound(TWOCOCYCLES[size]);
   if not known then
-    dir := DirectoriesPackageLibrary("rig", "cohomology")[1];
-    filename := Filename(dir, Concatenation("q", String(size), ".g"));
+    dir := DirectoriesPackageLibrary("rig", "homology")[1];
+    filename := Filename(dir, Concatenation("size", String(size), ".g"));
     if IsReadableFile(filename) then
       Read(filename);
     else
@@ -122,15 +122,15 @@ end);
 
 ### This function returns the generators of the generators of the 2nd quandle homology group of SmallQuandle(n, i)
 ### The database was computed with the function TorsionGenerators
-InstallGlobalFunction("2ndQuandleCohomologyGenerators"; function(size, number)
+InstallGlobalFunction("2ndQuandleCohomologyGenerators", function(size, number)
   local k, l, v, q, hom, gens;
 
   # If n is prime then the 2nd homology is trivial
   if IsPrime(size) then
     return rec( factors := [ ], generators := [ List([1..size^2], x->0) ]);
   fi;
 
-  hom := 2nd_quandlehomology_generators(size, number);
+  hom := 2ndQuandleHomologyGenerators(size, number);
   gens := [];
 
   for k in [1..Size(hom.generators)] do

init.g

@@ -11,4 +11,4 @@ ReadPackage("rig", "gap/affine.gd");
 ReadPackage("rig", "gap/derivations.gd");
 ReadPackage("rig", "knots/knots.gd");
 ReadPackage("rig", "upto9crossings.gd");
-#ReadPackage("rig", "homology/homology.gd");
+ReadPackage("rig", "homology/homology.gd");

knots/knots.gd

@@ -5,6 +5,8 @@ DeclareGlobalFunction("NrQuandleColorings");
 DeclareGlobalFunction("BoltzmannWeight");
 DeclareGlobalFunction("2CocycleInvariant");
 DeclareGlobalFunction("FundamentalQuandle");
+DeclareGlobalFunction("Signs");
 
 ### INTERNAL USE
+DeclareGlobalFunction("ReplacedList");
 DeclareGlobalFunction("CheckQuandleEquation");

knots/knots.gi

@@ -94,4 +94,83 @@ InstallGlobalFunction("2CocycleInvariant", function(fq, quandle, group, q)
   return sum;
 end);
 
+### This function replaces <old> by <new> in <list>
+InstallGlobalFunction("ReplacedList", function(list, old, new)
+  local j, r;
+  r := [];
+  for j in list do
+    if j = old then
+      Add(r, new);
+    else
+      Add(r, j);
+    fi;
+  od;
+  return r;
+end);
+
+
+### This function computes the fundamental quandle of a knot given by a planar diagram <pd>
+InstallGlobalFunction("FundamentalQuandle", function(pd)
+  local j, c, d, s, diagram, fq, f;
+
+  s := Signs(pd);
+  fq := [];
+
+  diagram := ShallowCopy(pd);
+
+  for c in diagram do
+    for j in [1..Size(diagram)] do
+      diagram[j] := ReplacedList(diagram[j], c[2], c[4]);
+    od;
+  od;
+
+  ### This is to normalize the arcs
+  f := Set(Flat(diagram));
+
+  for j in [1..Size(diagram)] do
+    if s[j] > 0 then
+      Add(fq, [Position(f, diagram[j][4]), Position(f, diagram[j][1]), Position(f, diagram[j][3]), 1]);
+    else
+      Add(fq, [Position(f, diagram[j][4]), Position(f, diagram[j][3]), Position(f, diagram[j][1]), -1]);
+    fi;
+  od;
+
+  return fq;
+end);
+
+### This function returns the signs of the crossings of a planar diagram <pd>
+### CHECK!
+InstallGlobalFunction("Signs", function(pd)
+  local c, s, max;
+
+  s := [];
+  max := Maximum(Flat(pd));
+
+  for c in pd do
+    if c[2] = max and c[4] = 1 then 
+      Add(s, 1);
+      continue;
+    elif c[2] = max and c[4] = max-1 then
+      Add(s, -1);
+      continue;
+    elif c[4] = max and c[2] = 1 then
+      Add(s, -1);
+      continue;
+    elif c[4] = max and c[2] = max-1 then
+      Add(s, 1);
+      continue;
+    elif c[4]>c[2] then
+      Add(s, 1);
+      continue;
+    elif c[4]<c[2] then
+      Add(s,-1);
+      continue;
+    else
+      Print("Nothing?\n");
+      return fail;
+    fi;
+  od;
+  return s;
+end);
+
 

read.g

@@ -10,7 +10,7 @@ ReadPackage("rig", "gap/extensions.gi");
 ReadPackage("rig", "gap/affine.gi");
 ReadPackage("rig", "knots/knots.gi");
 ReadPackage("rig", "knots/upto9crossings.g");
-#ReadPackage("rig", "homology/homology.gi");
+ReadPackage("rig", "homology/homology.gi");
 
 if LoadPackage("gbnp") <> fail then
   ReadPackage("rig", "gap/derivations.gi");