Pensieve header: The 1-co invariant for all ribbon knots with up to 10 crossings.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2016-09"];
<< OneSmidgen.m
tab=Get["tab.m"]

z[K_Knot] := (K/.tab)[[2]];
za[K_Knot] := z[K][[1]];
zp1[K_Knot] := Factor[((z[K][[4]] /. c|u|w->0)/((t-1)((za[K])^2)))];
zp2[K_Knot] := Factor[((Coefficient[z[K][[4]],c])/((t-1/t)(-4-t+(t^2))((za[K])^3)))];

Solve[-4-t+(t^2)==0]

Ribbons={(Subscript[6, 1]),(Subscript[8, 8]),(Subscript[8, 9]),(Subscript[8, 20]),(Subscript[9, 27]),(Subscript[9, 41]),(Subscript[9, 46]),(Subscript[10, 3]),(Subscript[10, 22]),(Subscript[10, 35]),(Subscript[10, 42]),(Subscript[10, 48]),(Subscript[10, 75]),(Subscript[10, 87]),(Subscript[10, 99]),(Subscript[10, 123]),(Subscript[10, 129]),(Subscript[10, 137]),(Subscript[10, 140]),(Subscript[10, 153]),(Subscript[10, 155])}/.(Subscript[n_, k_]):>Knot[n,k]

Length@Ribbons

Factor[Alexander[#][t]]& /@ Ribbons

MatrixForm@Table[{(Subscript[K[[1]], K[[2]]]), Factor@za[K], zp1[K], zp2[K], Vassiliev[2][K], Vassiliev[3][K]}, {K, Ribbons}]

MatrixForm@Table[{(Subscript[K[[1]], K[[2]]]), Factor@za[K], zp1[K], Factor[zp1[K]-(zp1[K]/.t->1/t)], zp2[K], Vassiliev[2][K], Vassiliev[3][K]}, {K, AllKnots[{3,8}]}]

MatrixForm@Table[{(Subscript[K[[1]], K[[2]]]), Factor@za[K], zp1[K], zp2[K], Vassiliev[2][K], Vassiliev[3][K]}, {K, Select[AllKnots[{3,10}], Vassiliev[3][#]==0&]}]

PolynomialGCD[(-1+2 t-(t^2)-(t^3)+2 (t^4)-(t^5)+(t^8)), (-1+(t^3)-2 (t^4)+(t^5)+(t^6)-2 (t^7)+(t^8))]

MatrixForm@Sort@Table[{Factor@za[K], zp1[K], zp2[K], (Subscript[K[[1]], K[[2]]])}, {K, AllKnots[{3,10}]}]

Length@Union@Table[Factor@za[K]->zp2[K], {K, AllKnots[{3,10}]}]

Length@Union@Table[{Factor@za[K]}, {K, AllKnots[{3,10}]}]

Union@Table[Expand@za[K]->Expand[-2zp2[K]], {K, AllKnots[{3,7}]}] // Column

Table[{(Subscript[K[[1]], K[[2]]]),Factor@za[K],Factor@(((za[K](t^2) D[za[K],t]+t(t-1)zp1[K])/(((t-1))^2))),2Exponent[za[K],t,Min]==Exponent[zp1[K],t,Min]∧2Exponent[za[K],t,Max]==Exponent[t zp1[K],t,Max], ThreeGenus[K]}, {K,Ribbons}] // MatrixForm