<< KnotTheory`

SubLink[pd_PD,js_List]:=Module[
{k,t0,t,t1,t2,S,P},t0=Flatten[List@@@Skeleton[pd][[js]]];
t=pd/.x_X:>Select[x,MemberQ[t0,#]&];
t=DeleteCases[t,X[]];
k=1;
While[
k≤Length[t],
If[Length[t[[k]]]<4,
t=Delete[t,k]/.(Rule@@t[[k]]),++k];
];
t1=List@@Union@@t;
t2=Thread[(t1)->Range[Length[t1]]];
S=t/.t2;
P=If[
S≠PD[]&&Length[S]≥2,
S,
PD[Knot[0,1]],
S
]
];
SubLink[pd_PD,j_]:=SubLink[pd,{j}];
SubLink[L_,js_]:=SubLink[PD[L],js];

Length[Ls=Select[AllLinks[{2,11}], Length[Skeleton[#]]==3&]]

PD[NewLink[1]] = PD[X[1,23,2,22],X[3,24,4,17],X[4,12,5,11],X[6,13,7,14],X[9,3,10,2],X[12,20,13,19],X[14,21,15,22],X[15,8,16,1],X[17,11,18,10],X[18,5,19,6],X[20,8,21,7],X[23,16,24,9]];
AppendTo[Ls, NewLink[1]];

trivials={-(1/(Sqrt[q]))-(Sqrt[q]), 1, -(1/(q^2))-(1/q), -q-q^2}

Bs=Select[Ls, And[
MemberQ[trivials, Jones[SubLink[#, {1,2}]][q]],
MemberQ[trivials, Jones[SubLink[#, {2,3}]][q]],
MemberQ[trivials, Jones[SubLink[#, {3,1}]][q]]
]&]

DrawMorseLink /@ Bs

DrawMorseLink[SubLink[Link[11,Alternating,434], #]]&  /@ {{1,2}, {2,3}, {3,1}}

MultivariableAlexander[SubLink[Link["L11a434"], #]][t]&  /@ {{1,2}, {2,3}, {3,1}}

trivialAs={0};
Select[Ls, And[
MemberQ[trivialAs, MultivariableAlexander[SubLink[#, {1,2}]][t]],
MemberQ[trivialAs, MultivariableAlexander[SubLink[#, {2,3}]][t]],
MemberQ[trivialAs, MultivariableAlexander[SubLink[#, {3,1}]][t]]
]&]