> # ``classicalR3Z3.mws'' > # This program computes the cocycle invariant for R_3 > # with Z_3 coefficients for the knot table. > # A 2-cocy is specified below by specific values and solutions. > clear: > with(LinearAlgebra): > > # Forming vectors. > f:=array(0..2,0..2,1..3): > > # Quandle cocycle conditions. > for i from 0 to 2 do > for k from 1 to 3 do > f[i,i,k]:=0 : > od: od: > > f[1,2,1]:=2: > f[1,2,2]:=2: > f[1,2,3]:=0: > f[1,0,1]:=0: > f[1,0,2]:=2: > f[1,0,3]:=0: > f[2,1,2]:=0: > ### The following are solutions found earlier, with the above substitution. > ### The above substitutions were found by earlier experiments. > f[2,0,1] := 1: > f[0,1,1] := 1: > f[0,2,1] := 0: > f[2,1,3] := 1: > f[0,1,2] := 2: > f[2,0,2] := 0: > f[0,1,3] := 1: > f[2,1,1] := 1: > f[0,2,2] := 2: > f[0,2,3] := 0: > f[2,0,3] := 0: > > h:=array(0..2,0..2): > for i from 0 to 2 do > for j from 0 to 2 do > h[i,j]:=Vector([f[i,j,1],f[i,j,2],f[i,j,3]]): > od: od: > > # Defining matrices, 1 = (2 3), 2 = (1 3), 0 = (1 2). > M[1]:=<<1,0,0>|<0,0,1>|<0,1,0>>: > M[2]:=<<0,0,1>|<0,1,0>|<1,0,0>>: > M[0]:=<<0,1,0>|<1,0,0>|<0,0,1>>: > > # Defining quandle operation. > for i from 0 to 2 do > for j from 0 to 2 do > g[i,j]:=2*j-i mod 3 : > od: od: > > # Defining 2-cocy equations. > for x from 0 to 2 do > for y from 0 to 2 do > for z from 0 to 2 do > E[x,y,z]:= M[z].h[x,y] + h[g[x,y],z] + M[g[g[x,y],z]].h[y,z] > - h[y,z] - M[g[y,z]].h[x,z] - h[g[x,z],g[y,z]]: > od: od: od: > > # Defining the set of equations. > EQ:=[]: > for x from 0 to 2 do > for y from 0 to 2 do > for z from 0 to 2 do > for w from 1 to 3 do > EQ:=[ op(EQ), E[x,y,z][w]=0 ] : > od: od: od: od: > Eq:=convert(EQ, set): # Converting the list EQ to a set Eq. > print(map( z -> z mod 3, Eq)); # This will be {0=0} if the given values are indeed solutions mod 3. > {0 = 0} > # Solving the cocycle conditions: > # Sol:=msolve(Eq, Var, 3); > > > # Braid words of the knot table, brind is the braid index. > # 3-colorable ones renumbered. > bw[1]:= [1,1,1]: brind[1]:=2: #3_1 > #bw[2]:= [1,-2,1,-2]: brind[2]:=3: #4_1 > #bw[3]:= [1,1,1,1,1]: brind[3]:=2: #5_1 > #bw[4]:= [1,1,2,2,-1,2]: brind[4]:=3: #5_2 > #bw[5] > bw[2]:= [-1,2,-1,3,-2,3,2]: brind[2]:=4: #brind[5]:=4: #6_1 > #bw[6]:= [-1,2,-1,2,2,2]: brind[6]:=3: > #bw[7]:= [-1,2,2,-1,-1,2]: brind[7]:=3: #6_3 > #bw[8]:= [1,1,1,1,1,1,1]: brind[8]:=2: #7_1 > #bw[9]:= [-1,3,3,3,2,1,1,-3,2]: brind[9]:=4: > #bw[10]:= [1,1,2,-1,2,2,2,2]: brind[10]:=3: > #bw[11] > bw[3]:= [1,1,2,3,3,-1,2,-3,2]: brind[3]:=4: > #bw[12]:= [1,1,1,1,2,-1,2,2]: brind[12]:=3: > #bw[13]:= [1,-2,-1,-1,3,2,2,2,3]: brind[13]:=4: > #bw[14] > bw[4]:= [1,-3,2,-3,2,-1,2,-3,2]: brind[4]:=4: #7_7 > #bw[15]:= [-1,2,3,-2,-1,4,4,3,2,-4]: brind[15]:=5: #8_1 > #bw[16]:= [-1,2,2,2,2,2,-1,2]: brind[16]:=3: > #bw[17]:= [-1,-1,-2,1,4,4,3,-4,-2,3]: brind[17]:=5: > #bw[18]:= [1,1,1,3,-2,-3,-3,1,-2]: brind[18]:=4: > #bw[19] > bw[5]:= [1,1,1,-2,1,1,1,-2]: brind[5]:=3: #8_5 > #bw[20]:= [-1,2,-1,-3,2,2,2,3,3]: brind[20]:=4: > #bw[21]:= [1,1,1,1,-2,-2,1,-2]: brind[21]:=3: > #bw[22]:= [-1,2,1,1,-3,2,2,-3,-3]: brind[22]:=4: > #bw[23]:= [-1,2,-1,-1,-1,2,2,2]: brind[23]:=3: > #bw[24] > bw[6]:= [-1,2,2,-1,-1,2,2,2]: brind[6]:=3: #8_10 > #bw[25] > bw[7]:= [-1,2,2,-3,2,3,3,-1,2]: brind[7]:=4: > #bw[26]:= [1,-2,3,-4,3,-4,2,1,-3,-2]: brind[26]:=5: > #bw[27]:= [1,1,2,-3,2,-1,-3,-3,2]: brind[27]:=4: > #bw[28]:= [1,1,2,2,-1,-3,2,-3,2]: brind[28]:=4: > #bw[29] > bw[8]:= [1,1,-2,1,3,3,2,2,3]: brind[8]:=4: #8_15 > #bw[30]:= [1,1,-2,1,1,-2,1,-2]: brind[30]:=3: > #bw[31]:= [-1,2,-1,2,2,-1,-1,2]: brind[31]:=3: > #bw[32] > bw[9]:= [1,-2,1,-2,1,-2,1,-2]: brind[9]:=3: > #bw[33] > bw[10]:= [1,2,1,2,1,2,2,1]: brind[10]:=3: > #bw[34] > bw[11]:= [1,1,1,2,-1,-1,-1,2]: brind[11]:=3: #8_20 > #bw[35] > bw[12]:= [1,-2,-2,1,1,2,2,2]: brind[12]:=3: #8_21 > #bw[36] > bw[13]:= [1,1,1,1,1,1,1,1,1]: brind[13]:=2: #9_1 > #bw[37] > bw[14]:= [1,2,3,4,4,4,3,-4,2,1,-3,-2]: brind[14]:=5: > #bw[38]:= [1,-2,1,1,1,1,1,1,2,2]: brind[38]:=3: > #bw[39] > bw[15]:= [-1,3,2,1,1,2,3,3,3,3,-2]: brind[15]:=4: > #bw[40]:= [1,1,2,-1,3,-2,3,4,4,3,-4,2]: brind[40]:=5: #9_5 > #bw[41] > bw[16]:= [1,1,2,2,1,1,1,1,1,-2]: brind[16]:=3: > #bw[42]:= [1,1,1,2,3,3,-1,2,2,2,-3]: brind[42]:=4: > #bw[43]:= [1,-2,3,-1,-1,-4,3,-2,3,3,4,3]: brind[43]:=5: > #bw[44]:= [1,1,1,-2,1,1,1,1,2,2]: brind[44]:=3: > #bw[45] > bw[17]:= [-1,2,1,1,2,2,2,-3,2,3,3]: brind[17]:=4: #9_10 > #bw[46] > bw[18]:= [-1,2,-3,2,-1,2,2,2,2,3,2]: brind[18]:=4: > #bw[47]:= [1,-2,-1,-1,3,2,2,2,4,4,3,-4]: brind[47]:=5: > #bw[48]:= [1,1,2,-3,2,-1,3,3,2,2,2]: brind[48]:=4: > #bw[49]:= [1,4,4,-3,2,-3,2,-3,-1,-4,2,2,3,-2]: brind[49]:=5: > #bw[50] > bw[19]:= [1,-2,1,3,-2,4,-3,4,4,3]: brind[19]:=5: #9_15 > #bw[51] > bw[20]:= [1,1,-2,1,1,1,2,2,2,2]: brind[20]:=3: > #bw[52] > bw[21]:= [1,1,1,3,-2,1,-2,-3,-2,1,-2]: brind[21]:=4: > #bw[53]:= [1,1,-3,2,-1,2,2,3,3,2,2]: brind[53]:=4: > #bw[54]:= [1,1,2,-1,-3,-4,-3,2,-3,2,4,-3]: brind[54]:=5: > #bw[55]:= [1,2,2,-3,2,-1,2,-3,2,2,2]: brind[55]:=4: #9_20 > #bw[56]:= [-1,-1,3,-4,3,-2,1,3,4,4,2,2]: brind[56]:=5: > #bw[57]:= [1,-2,3,3,3,-2,3,-1,-2,3,-2]: brind[57]:=4: > #bw[58] > bw[22]:= [1,-2,1,1,2,3,3,3,2,-3,2]: brind[22]:=4: > #bw[59] > bw[23]:= [1,3,3,-2,1,3,-2,-2,-2]: brind[23]:=4: > #bw[60]:= [-1,2,-1,-4,-3,2,4,4,3,4,4,2,2,-3]: brind[60]:=5: #9_25 > #bw[61]:= [-1,2,2,2,3,2,-1,2,-1,-3,2]: brind[61]:=4: > #bw[62]:= [-1,2,-1,-1,-3,2,-1,2,2,3,2]: brind[62]:=4: > #bw[63] > bw[24]:= [1,1,3,3,-2,-2,1,3,-2]: brind[24]:=4: > #bw[64] > bw[25]:= [-1,2,-3,2,-1,2,-3,2,2]: brind[25]:=4: > #bw[65]:= [1,-3,-3,2,-3,-1,2,1,1,-3,2]: brind[65]:=4: #9_30 > #bw[66]:= [-1,2,2,-3,2,2,-1,2,3,-1,2]: brind[66]:=4: > #bw[67]:= [-1,2,3,-1,2,-1,3,3,2,-3,2]: brind[67]:=4: > #bw[68]:= [-1,-1,-1,2,-1,2,2,3,1,-2,3]: brind[68]:=4: > #bw[69] > bw[26]:= [1,-2,3,-2,1,-2,3,1,-2]: brind[26]:=4: > #bw[70] > bw[27]:= [1,3,3,-4,-2,1,2,3,3,2,-3,4,-3,2]: brind[27]:=5: #9_35 > #bw[71]:= [-1,2,2,2,-3,2,3,3,-1,2,3]: brind[71]:=4: > #bw[72] > bw[28]:= [1,-2,3,-2,3,-1,4,-3,-2,4,3,-2]: brind[28]:=5: > #bw[73] > bw[29]:= [1,1,2,3,3,2,2,-1,2,-3,2]: brind[29]:=4: > #bw[74]:= [-1,-3,2,4,3,3,-1,2,2,3,4,-2]: brind[74]:=5: > #bw[75] > bw[30]:= [1,3,-2,3,1,-2,1,3,-2]: brind[30]:=4: #9_40 > #bw[76]:= [-1,2,2,-4,3,-4,-2,1,-3,2,3,-4,3,2]: brind[76]:=5: > #bw[77]:= [1,1,1,3,-2,3,-1,-1,-2]: brind[77]:=4: > #bw[78]:= [1,2,1,1,2,2,3,-2,1,-2,-3]: brind[78]:=4: > #bw[79]:= [-1,2,-1,3,-2,3,2,2,-3]: brind[79]:=4: > #bw[80]:= [1,-2,1,3,2,2,2,3,-2]: brind[80]:=4: #9_45 > #bw[81] > bw[31]:= [1,3,2,-1,-3,2,1,3,-2]: brind[31]:=4: > #bw[82] > bw[32]:= [-1,2,3,-1,2,-1,2,3,2]: brind[32]:=4: > #bw[83] > bw[33]:= [1,1,-2,3,2,2,-1,-3,2,-3,2]: brind[33]:=4: > #bw[84]:= [1,1,2,2,3,2,-1,2,2,3,-2]: brind[84]:=4: #9_49 > > > for KT from 1 to 33 do # KT is the number in the Knot Table. Up to 84. > > print(Knot, KT); > > # Setting up the state-sum terms. > SST:= [ ] : > > # Defining Burau rep for R_3, acting from the right to row vectors. > Burau:=linalg[matrix](2,2,[0,-1,1,2]): > Burinv:=linalg[matrix](2,2,[2,1,-1,0]): > > for j0 from 1 to brind[KT]-1 do # Copying the Burau matrix into bigger ones. > B[j0]:=array(1..brind[KT],1..brind[KT]): > B[-j0]:=array(1..brind[KT],1..brind[KT]): > od: > ID:=Matrix(1..brind[KT],1..brind[KT],shape=identity); > for j1 from 1 to brind[KT]-1 do > for j2 from 1 to brind[KT] do > for j3 from 1 to brind[KT] do > B[j1][j2,j3]:=linalg[copyinto](Burau, ID, j1,j1)[j2,j3]: > B[-j1][j2,j3]:=linalg[copyinto](Burinv, ID, j1,j1)[j2,j3]: > od: od: od: > > # Color vectors. > for jj3 from 1 to (nops(bw[KT])+1) do > Color[jj3]:=array(1..brind[KT]): > od: > > # Producing all color vectors. > num:=3^(brind[KT]) : > for indx from 0 to (num-1) do # One color at a time. > > for jj5 from 1 to brind[KT] do > Color[1][jj5]:=iquo(indx,3^(jj5-1)) mod 3: > od: > > # Computing all color vectors. > for jj6 from 1 to nops(bw[KT]) do > Newcolorvec[jj6]:=evalm(Color[jj6] &* B[bw[KT][jj6]]): > for jj8 from 1 to brind[KT] do > Color[jj6+1][jj8]:=map( z -> z mod 3, Newcolorvec[jj6][jj8]): > od: > od: > > SSTcontri:=Vector([0,0,0]): > # State-sum contributions. > > # Finding if the colors match. > ColorDiff0:=evalm(Color[1]-Color[nops(bw[KT])+1]); > ColDiffMatch0:=sum(abs(ColorDiff0[jj]),jj=1..brind[KT]); > # This is zero iff the top color vec matches the bottom. > > > if ColDiffMatch0 =0 then > > > for s from 1 to nops(bw[KT]) do > > # Computing the weight in front of the cocycle. > > WT[0]:=Matrix(1..3,1..3,shape=identity): > WT[1]:=M[Color[s][brind[KT]]]: > # WT[1] is the weight for the crossing of \sigma_{n-2}. The target region is > # to the right of the crossing. > for k7 from 1 to (brind[KT]-1) do > WT[k7+1]:=MatrixMatrixMultiply( WT[k7], M[Color[s][brind[KT]-k7]] ) : > od: > > # The weight WGT[1] for the \signa_1 is WT[n-2], > # and WGT[2]=WT[n-3], ... , WGT[n-2]=WT[1]. > > if brind[KT]>2 then > for k77 from 1 to (brind[KT]-2) do > WGT[k77]:=WT[brind[KT]-k77-1]: > od: > WGT[brind[KT]-1]:=WT[0]: > else > WGT[1]:=WT[0]: > fi: > > if (bw[KT][s])>0 then > > SSTcontri:=evalm( SSTcontri + > MatrixVectorMultiply(WGT[abs(bw[KT][s])], > h[Color[s][abs(bw[KT][s])], Color[s][abs(bw[KT][s])+1] ] ) ): > > else > > SSTcontri:=evalm( SSTcontri - > MatrixVectorMultiply(WGT[abs(bw[KT][s])], > h[Color[s+1][abs(bw[KT][s])], Color[s][abs(bw[KT][s])] ] ) ): > > fi: > > od: # Closing the state-sum term, for s. > > #SST:=[ op(SST), map(z -> z mod 3, subs(Sol, evalm(SSTcontri) ) ) ]: > SST:=[ op(SST), map(z -> z mod 3, evalm(SSTcontri) ) ]: > > fi: # Closing the case when it colors. > > > od: # Closing one color here at a time, for indx. > > > print(SST); > > od: # Closing KT. > Knot, 1 [[0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0]] Knot, 2 [[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]] Knot, 3 [[0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0]] Knot, 4 [[0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0]] Knot, 5 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 6 [[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]] Knot, 7 [[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]] Knot, 8 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 9 [[0, 0, 0], [1, 1, 1], [1, 1, 1], [2, 2, 2], [2, 2, 2], [1, 1, 1], [2, 2, 2], [1, 1, 1], [2, 2, 2], [2, 2, 2], [2, 2, 2], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [2, 2, 2], [2, 2, 2], [2, 2, 2], [1, 1, 1], [2, 2, 2], [1, 1, 1], [2, 2, 2], [2, 2, 2], [1, 1, 1], [1, 1, 1], [0, 0, 0]] Knot, 10 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 11 [[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]] Knot, 12 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 13 [[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]] Knot, 14 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 15 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 16 [[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]] Knot, 17 [[0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0]] Knot, 18 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 19 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 20 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 21 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 22 [[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]] Knot, 23 [[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]] Knot, 24 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 25 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 26 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 27 [[0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [0, 0, 0], [0, 0, 0]] Knot, 28 [[0, 0, 0], [1, 1, 1], [1, 1, 1], [0, 0, 0], [0, 0, 0], [2, 2, 2], [0, 0, 0], [2, 2, 2], [0, 0, 0], [0, 0, 0], [0, 0, 0], [2, 2, 2], [1, 1, 1], [0, 0, 0], [1, 1, 1], [2, 2, 2], [0, 0, 0], [0, 0, 0], [0, 0, 0], [2, 2, 2], [0, 0, 0], [2, 2, 2], [0, 0, 0], [0, 0, 0], [1, 1, 1], [1, 1, 1], [0, 0, 0]] Knot, 29 [[0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0]] Knot, 30 [[0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0], [2, 2, 2], [2, 2, 2], [2, 2, 2], [0, 0, 0]] Knot, 31 [[0, 0, 0], [0, 0, 0], [0, 0, 0], [2, 2, 2], [1, 1, 1], [0, 0, 0], [2, 2, 2], [0, 0, 0], [1, 1, 1], [1, 1, 1], [2, 2, 2], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [2, 2, 2], [1, 1, 1], [1, 1, 1], [0, 0, 0], [2, 2, 2], [0, 0, 0], [1, 1, 1], [2, 2, 2], [0, 0, 0], [0, 0, 0], [0, 0, 0]] Knot, 32 [[0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [0, 0, 0], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0]] Knot, 33 [[0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0], [1, 1, 1], [1, 1, 1], [1, 1, 1], [0, 0, 0]] > > >