Pensieve header:  With Joe Boninger: $\rho_1$ is not log-concave even on alternating knots. Continues pensieve://2023-03/ (for Alexander).

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\APAI"];
Once[<<KnotTheory`; <<Rot.m];

ConcaveQ[s_List]:=And@@Thread[2s[[2;;-2]]>=s[[1;;-3]]+s[[3;;-1]]];

AlexanderLogConcaveQ[K_]:=Module[{alex,T,cs},
alex=Alexander[K][-T];
cs=CoefficientList[Sign[alex/.T->1](T^(Exponent[alex,T]))alex,T];
If[Not[And@@(Positive/@cs)],Return@False];
If[Length[cs]<3, Return@True];
ConcaveQ[Log@cs]
]

NLCs=Select[AllKnots[{3,10}], Not[AlexanderLogConcaveQ[#]]&]

AlternatingQ/@NLCs

Total[
AlternatingQ/@Select[AllKnots[{3,10}], Not[AlexanderLogConcaveQ[#]]&]
]

(Subscript[ρ, 1])

(Subscript[R, 1])[s_,i_,j_]:=s((Subscript[g, j,i])((Subscript[g, (SuperPlus[j]),j])+(Subscript[g, j,(SuperPlus[j])])-(Subscript[g, i,j]))-(Subscript[g, i,i])((Subscript[g, j,(SuperPlus[j])])-1)-1/2);
ρ[K_]:=ρ[K]=Module[{Cs,φ,n,A,s,i,j,k,Δ,G,ρ1},
{Cs,φ}=Rot[K]; n=Length[Cs];
A=IdentityMatrix[2n+1];
Cases[Cs,{s_,i_,j_}:>(A[[{i,j},{i+1,j+1}]]+=(({{-(T^s), (T^s)-1}, {0, -1}})))];
Δ=(T^((-Total[φ]-Total[Cs[[All,1]]])/2))Det[A];
G=Inverse[A];
ρ1=(∑_(k=1)^n)(Subscript[R, 1])@@Cs[[k]] - (∑_(k=1)^(2n))φ[[k]]((Subscript[g, k,k])-1/2);
Factor@{Δ,(Δ^2)ρ1/.(SuperPlus[α_]):>α+1/.(Subscript[g, α_,β_]):>G[[α,β]]}];

K=Knot[8,4]

ρ[K][[2]]//Factor

ρ1=Expand@Factor[((T ρ[K][[2]])/((((-1+T))^2) ))/.T->-T]

cs=CoefficientList[Sign[ρ1/.T->1](T^(Exponent[ρ1,T]))ρ1,T]

ConcaveQ[Log@cs]

ρ1LogConcaveQ[K_]:=Module[{ρ1,cs},
ρ1=Expand[ρ[K][[2]]/.T->-T];
cs=CoefficientList[Sign[ρ1/.T->1](T^(Exponent[ρ1,T]))ρ1,T];
If[Not[And@@(Positive/@cs)],Return@False];
If[Length[cs]<3, Return@True];
ConcaveQ[Log@cs]
]

NLCs=Select[AllKnots[{3,10}], Not[ρ1LogConcaveQ[#]]&]

AlternatingQ/@NLCs

Total[
AlternatingQ/@Select[AllKnots[{3,10}], Not[AlexanderLogConcaveQ[#]]&]
]