Pensieve header: Analyzing the 1-smidgen invariant on the Rolfsen table.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2016-09"];
<<KnotTheory`
tab=Get["tab.m"]⋃Get["tab11.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)))];

Column[AllKnots[{3,7}] /. tab /. {_,𝔼[a_,0,0,p_]} :> {a,p/.{c|u|w->0}, Coefficient[p, c], Coefficient[p,u w]}]

(1/(((9+(1/(t^2))-(5/t)-5 t+(t^2)))^3))(-10144+(4/(t^8))-(65/(t^7))+(949/(2 (t^6)))-(4037/(2 (t^5)))+(10819/(2 (t^4)))-(18121/(2 (t^3)))+(16217/(2 (t^2)))+(247/t)+12406 t-6608 (t^2)+191 (t^3)+((4279 (t^4))/2)-((3131 (t^5))/2)+((1193 (t^6))/2)-((271 (t^7))/2)+((35 (t^8))/2)-(t^9))//Simplify

PolynomialRemainder[-10144+(4/(t^8))-(65/(t^7))+(949/(2 (t^6)))-(4037/(2 (t^5)))+(10819/(2 (t^4)))-(18121/(2 (t^3)))+(16217/(2 (t^2)))+(247/t)+12406 t-6608 (t^2)+191 (t^3)+((4279 (t^4))/2)-((3131 (t^5))/2)+((1193 (t^6))/2)-((271 (t^7))/2)+((35 (t^8))/2)-(t^9), 9+(1/(t^2))-(5/t)-5 t+(t^2),t]

Column[AllKnots[{3,7}] /. tab /. {_,𝔼[a_,0,0,p_]} :> {a,Factor[((p/.{c|u|w->0})/((t-1)(a^2)))], Factor[((Coefficient[p, c])/(((t^2)-1)(-4-t+(t^2)) (a^3)))], Simplify[(((t-1)Coefficient[p,u w]-Coefficient[p, c])/((1+t) (-4-t+(t^2))(a^3)))]}]

mat=AllKnots[{3,8}] /. tab /. {_,𝔼[a_,0,0,p_]} :> {a,Factor[((p/.{c|u|w->0})/((t-1)(a^2)))], Factor[((Coefficient[p, c])/(((t^2)-1)(-4-t+(t^2)) (a^3)))], Simplify[(((t-1)Coefficient[p,u w]-Coefficient[p, c])/((1+t) (-4-t+(t^2))(a^3)))]}

mat // Dimensions

MatrixRank[mat /. t->RandomReal[] /. {x_,y_,z_,0} :> {1,x,y,z, (x^2),x y,x z,(y^2),(z^2),y z}]

MatrixRank[mat/. {x_,y_,z_,0} :> ({1,x,y,z, t,t x,t y,t z}/. t->RandomReal[])  ]

MatrixRank[mat/. {x_,y_,z_,0} :> ({1,x,y,z, t,t x,t y,t z, (t^2),(t^2) x,((t )^2)y,(t^2) z}/. t->RandomReal[])  ]

MatrixRank[mat /. t->RandomReal[] /. {x_,y_,z_,0} :> {1,x,y,z,(x^2),x y,x z,(x^3),(x^2) y,(x^2) z,(x^4),(x^3) y,(x^3) z}]

MatrixRank[mat /. t->RandomReal[] /. {x_,y_,z_,0} :> {1,x,y,z,(x^2),x y,x z,(x^3),(x^2) y,(x^2) z,(x^4),(x^3) y,(x^3) z,(x^5),(x^4) y,(x^4) z}]

DeleteCases[Table[K->Together[((t D[za[K],t])/(t-(t^(-1))))+2zp2[K]], {K,AllKnots[{3,11}]}], _->0]

Table[{(Subscript[K[[1]], K[[2]]]),Factor@za[K],zp1[K]}, {K,AllKnots[{3,8}]}] // MatrixForm

Table[{(Subscript[K[[1]], K[[2]]]),e1=Exponent[za[K],t,Min],e2=Exponent[zp1[K],t,Min], e2==2e1}, {K,AllKnots[{3,7}]}] // MatrixForm

Select[AllKnots[{3,11}], 2Exponent[za[#],t,Min]!=Exponent[zp1[#],t,Min]&]

Table[{(Subscript[K[[1]], K[[2]]]),e1=Exponent[za[K],t,Max],e2=Exponent[zp1[K],t,Max], e2==2e1-1, e3=ThreeGenus[K], e1==e3}, {K,AllKnots[{3,8}]}] // MatrixForm

Select[AllKnots[{3,11}], 2Exponent[za[#],t,Max]-1!=Exponent[zp1[#],t,Max]&]

gna=Select[AllKnots[{3,11}], Exponent[za[#],t,Max]≠ThreeGenus[#]&]

ThreeGenus /@ gna

Exponent[za[#],t,Max]& /@ gna

Exponent[zp1[#],t,Min]& /@ gna

Exponent[zp1[#],t,Max]& /@ gna

ThreeGenus /@ AllKnots[12]

?ThreeGenus

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,AllKnots[{3,8}]}] // MatrixForm

Table[{(Subscript[K[[1]], K[[3]]]),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,{Knot[11,NonAlternating,34],Knot[11,NonAlternating,42],Knot[11,NonAlternating,45],Knot[11,NonAlternating,67],Knot[11,NonAlternating,73],Knot[11,NonAlternating,97],Knot[11,NonAlternating,152]}}] // MatrixForm

Table[{(Subscript[K[[1]], K[[2]]]),ExpandNumerator@Together@((za[K])^2),ExpandNumerator@Together@(za[K]t D[za[K],t]+(t-1)zp1[K]),2Exponent[za[K],t,Min]==Exponent[zp1[K],t,Min]∧2Exponent[za[K],t,Max]==Exponent[t zp1[K],t,Max], ThreeGenus[K]}, {K,AllKnots[{3,8}]}] // MatrixForm

Table[{za[K],K,Factor@(((za[K](t^2) D[za[K],t]+t(t-1)zp1[K])/(((t-1))^2)))}, {K, AllKnots[{3,10}]}]//Sort // MatrixForm

Table[{Expand@Factor@(((za[K](t^2) D[za[K],t]+t(t-1)zp1[K])/(((t-1))^2))),za[K],K, SymmetryType[K]}, {K, AllKnots[{3,7}]}] // MatrixForm

Table[{Factor@(((za[K](t^2) D[za[K],t]+t(t-1)zp1[K])/(((t-1))^2))),za[K],K, SymmetryType[K]}, {K, AllKnots[{3,10}]}]//Sort // MatrixForm