D1=PD[X[1,5,2,4],X[5,3,6,2],X[3,1,4,6]];
D2=PD[X[4,1,5,2],X[2,5,3,6],X[6,3,1,4]];
K41=PD[X[4,2,5,1],X[8,6,1,5],X[6,3,7,4],X[2,7,3,8]];
ConwayK=PD[X[4,2,5,1],X[8,4,9,3],X[12,5,13,6],X[2,8,3,7],X[9,17,10,16],X[11,18,12,19],X[6,13,7,14],X[15,20,16,21],X[17,1,18,22],X[19,14,20,15],X[21,10,22,11]];
PK=PD[
X[4,48,5,47],X[5,40,6,41],X[6,68,7,67],X[11,97,12,96],
X[12,62,13,61],X[13,26,14,27],X[14,82,15,81],X[20,105,21,106],
X[22,43,23,44],X[23,65,24,64],X[24,9,25,10],X[25,99,26,98],
X[29,18,30,19],X[30,90,31,89],X[31,2,32,3],X[32,76,33,75],
X[35,53,36,52],X[36,72,37,71],X[39,49,40,48],X[44,21,45,22],
X[45,87,46,86],X[50,38,51,37],X[51,70,52,71],X[54,34,55,33],
X[55,74,56,75],X[56,4,57,3],X[57,88,58,89],X[58,20,59,19],
X[62,97,63,98],X[63,11,64,10],X[68,49,69,50],X[69,39,70,38],
X[72,53,73,54],X[73,35,74,34],X[76,2,77,1],X[77,90,78,91],
X[78,18,79,17],X[82,99,83,100],X[83,9,84,8],X[84,65,85,66],
X[85,43,86,42],X[87,105,88,104],X[92,16,93,15],X[93,80,94,81],
X[94,28,95,27],X[95,60,96,61],X[100,8,101,7],X[101,66,102,67],
X[102,42,103,41],X[103,46,104,47],X[106,60,107,59],X[107,28,108,29],
X[108,80,109,79],X[109,16,110,17],X[110,92,1,91]
];

KhSize[pd_]:=Module[{w,front,TL,todo,v,x,t1,t2,t3,t4},
w=Plus@@(pd/.{
X[i_,j_,k_,l_]/;j-l==1∨l-j>1:>1,
X[i_,j_,k_,l_]/;l-j==1∨j-l>1:>-1
});
SetAttributes[p,Orderless];
front={};
TL=1;
todo=List@@pd;
v[x_]:=Length[front⋂List@@x];
While[Length[todo]>0,
x=RandomChoice[MaximalBy[todo,v]];
t1=TL(x/.X[i_,j_,k_,l_]:>p[i,j]p[k,l]+ p[i,l]p[j,k]);
t2=Expand[t1];
t3=t2//.p[i_,j_]p[j_,k_]:>p[i,k];
TL=t3/.{p[i_,i_]:>2, ((p[i_,j_])^2):>2};
todo=Complement[todo,{x}];
front=Complement[front⋃List@@x, front⋂List@@x]
];
TL
]

KhSize/@{D1,D2,K41,ConwayK,PK}

Length[PK]

(62817105453162044382^(1/55))//N