"Relation 1 "
AT[{Pos[1,2,3,4]},0,2,{{1,3,2,4}}]-AT[{Neg[1,2,3,4]},0,2,{{1,3,2,4}}]===Expand[(t[1]-1)*AT[{P[4,1],P[3,2]},0,2,{{1,3,2,4}} ]]

"Relation 2"

AT[{CD[1,3,2,4]},0,2,{{1,3,2,4}}]===Expand[t[1]*AT[{P[1,4],P[2,3]},0,2,{{1,3,2,4}}]]

"Relation 6 - 4 Y"

AT[{CD[2,1,4,8],CD[5,9,1,6],HBlob[3,7]},1,5,{{3,7},{8,4},{9,5},{6,1,2}}]-AT[{CD[5,1,3,7],CD[4,8,1,9],HBlob[2,6]},1,5,{{3,7},{8,4},{9,1,5},{6,2}}]+AT[{CD[4,1,2,6],CD[3,7,1,8],HBlob[5,9]},1,5,{{3,7},{8,1,4},{9,5},{6,2}}]-AT[{CD[3,1,5,9],CD[2,6,1,7],HBlob[4,8]},1,5,{{3,1,7},{8,4},{9,5},{6,2}}]-(AT[{CD[2,1,5,9],CD[4,8,1,6],HBlob[3,7]},1,5,{{3,7},{8,4},{9,5},{6,1,2}}]-AT[{CD[5,1,4,8],CD[3,7,1,9],HBlob[2,6]},1,5,{{3,7},{8,4},{9,1,5},{6,2}}]+AT[{CD[4,1,3,7],CD[2,6,1,8],HBlob[5,9]},1,5,{{3,7},{8,1,4},{9,5},{6,2}}]-AT[{CD[3,1,2,6],CD[5,9,1,7],HBlob[4,8]},1,5,{{3,1,7},{8,4},{9,5},{6,2}}])

"Relation 7 - Welded Knots"
AT[{Pos[7,1,3,6],Pos[3,2,4,5],P[1,9],P[2,8]},3,6,{{5,2,8},{6,1,9},{7,3,4}}]===AT[{Pos[7,8,1,3],Pos[1,9,4,2],P[5,3],P[6,2]},3,6,{{5,3,8},{6,2,9},{7,1,4}}]

"Relation 9 - Virtual Tails Commute"

AT[{Arr[3,7,2,1],Arr[4,5,1,6]},1,4,{{1,2,6},{3,7},{4,5}}]===AT[{Arr[3,7,1,6],Arr[4,5,2,1]},1,4,{{1,2,6},{3,7},{4,5}}]

"Relation 13 - No internal vertices for arrow diagrams"

AT[{Arr[1,9,8,13],Arr[2,1,7,12],Arr[3,2,6,11],Arr[4,3,5,10]},3,8,{{1,2,3,4,9},{8,13},{7,12},{6,11},{5,10}}]-AT[{Arr[1,9,7,12],Arr[2,1,8,13],Arr[3,2,6,11],Arr[4,3,5,10]},3,8,{{1,2,3,4,9},{8,13},{7,12},{6,11},{5,10}}]-AT[{Arr[1,9,8,13],Arr[2,1,7,12],Arr[3,2,5,10],Arr[4,3,6,11]},3,8,{{1,2,3,4,9},{8,13},{7,12},{6,11},{5,10}}]+AT[{Arr[1,9,7,12],Arr[2,1,8,13],Arr[3,2,5,10],Arr[4,3,6,11]},3,8,{{1,2,3,4,9},{8,13},{7,12},{6,11},{5,10}}]-(AT[{Arr[1,9,6,11],Arr[2,1,5,10],Arr[3,2,8,13],Arr[4,3,7,12]},3,8,{{1,2,3,4,9},{8,13},{7,12},{6,11},{5,10}}]-AT[{Arr[1,9,5,10],Arr[2,1,6,11],Arr[3,2,8,13],Arr[4,3,7,12]},3,8,{{1,2,3,4,9},{8,13},{7,12},{6,11},{5,10}}]-AT[{Arr[1,9,6,11],Arr[2,1,5,10],Arr[3,2,7,12],Arr[4,3,8,13]},3,8,{{1,2,3,4,9},{8,13},{7,12},{6,11},{5,10}}]+AT[{Arr[1,9,5,10],Arr[2,1,6,11],Arr[3,2,7,12],Arr[4,3,8,13]},3,8,{{1,2,3,4,9},{8,13},{7,12},{6,11},{5,10}}])

"Alternate Version for 13 - Not 0!"
AT[{Arr[1,9,8,13],Arr[2,14,7,12],Arr[3,15,6,11],Arr[4,16,5,10]},0,8,{{1,9},{2,14},{3,15},{4,16},{8,13},{7,12},{6,11},{5,10}}]-AT[{Arr[1,9,7,12],Arr[2,14,8,13],Arr[3,15,6,11],Arr[4,16,5,10]},0,8,{{1,9},{2,14},{3,15},{4,16},{8,13},{7,12},{6,11},{5,10}}]-AT[{Arr[1,9,8,13],Arr[2,14,7,12],Arr[3,15,5,10],Arr[4,16,6,11]},0,8,{{1,9},{2,14},{3,15},{4,16},{8,13},{7,12},{6,11},{5,10}}]+AT[{Arr[1,9,7,12],Arr[2,14,8,13],Arr[3,15,5,10],Arr[4,16,6,11]},0,8,{{1,9},{2,14},{3,15},{4,16},{8,13},{7,12},{6,11},{5,10}}]-(AT[{Arr[1,9,6,11],Arr[2,14,5,10],Arr[3,15,8,13],Arr[4,16,7,12]},0,8,{{1,9},{2,14},{3,15},{4,16},{8,13},{7,12},{6,11},{5,10}}]-AT[{Arr[1,9,5,10],Arr[2,14,6,11],Arr[3,15,8,13],Arr[4,16,7,12]},0,8,{{1,9},{2,14},{3,15},{4,16},{8,13},{7,12},{6,11},{5,10}}]-AT[{Arr[1,9,6,11],Arr[2,14,5,10],Arr[3,15,7,12],Arr[4,16,8,13]},0,8,{{1,9},{2,14},{3,15},{4,16},{8,13},{7,12},{6,11},{5,10}}]+AT[{Arr[1,9,5,10],Arr[2,14,6,11],Arr[3,15,7,12],Arr[4,16,8,13]},0,8,{{1,9},{2,14},{3,15},{4,16},{8,13},{7,12},{6,11},{5,10}}])===0

"The double delta move - Stanford-Naik - Relation 15 - can't compute run out of memory"

t[1]=x; t[2]=x; t[3]=x; t[4]=x; t[5]=x; t[6]=x;

AT[{Pos[27,7,6,1],Pos[28,29,5,7],Pos[5,30,4,8],Pos[6,8,3,2],Pos[11,4,19,15],Pos[10,3,11,14],Pos[18,15,20,21],Pos[17,14,15,18],Neg[26,1,9,12],Neg[9,2,10,13],Neg[16,13,17,23],Neg[25,12,16,24]},18,24,{{19,11,10,9,26},{20,18,17,16,25},{21,15,4,5,28},{22,14,3,6,27},{23,13,2,8,30},{24,12,1,7,29}}]===AT[{Pos[26,27,1,4],Pos[1,28,2,5],Pos[8,5,9,12],Pos[25,4,8,11],Neg[2,29,3,6],Neg[3,30,19,7],Neg[10,7,20,14],Neg[9,6,10,13],Pos[12,13,16,15],Pos[16,14,21,17],Pos[11,15,18,24],Pos[18,17,22,23]},18,24,{{19,26,1,2,3},{25,8,9,10,20},{27,4,11,18,22},{28,5,12,16,21},{29,6,13,15,24},{30,7,14,17,23}}]

"What about Welded knots?"
AT[{Neg[9,10,1,6],Neg[1,11,2,5],Neg[2,12,3,8],Neg[3,13,4,7]},3,8,{{9,1,2,3,4},{10,6},{5,11},{12,8},{7,13}}]===AT[{Neg[9,12,1,8],Neg[1,13,2,7],Neg[2,10,3,6],Neg[3,11,4,5]},3,8,{{9,1,2,3,4},{10,6},{5,11},{12,8},{7,13}}]