Identifying Knots within a List

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IdentifyWithin[L,H], whose code is available here, returns those elements from the list of knots $H$, whose invariant matches that of the knot $L$. It can also recognize mirrors and connected sums of the knots in the list. Its options include turning off (on) the search for connected sums with ConnectedSum->False (True) and choosing the invariants to be used in identification by selecting, for example, Invariants->{Jones[#][q]&, HOMFLYPT[#][a,z]&}. IdentifyWithin can be used together with SubLink to determine the components of a link. For the second component of link L11n150, for instance, we get:

(For In[1] see Setup)

In[2]:=

Import["http://katlas.org/wiki/IdentifyWithin.m&action=raw"];

In[3]:=

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] >= 3, S, PD[Knot[0, 1]], S] ]; SubLink[pd_PD, j_] := SubLink[pd, {j}]; SubLink[L_, js_] := SubLink[PD[L], js];

In[4]:=	IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]
Out[4]=	{Knot[5, 2]}

L11n150

5_2

Unfortunately, the program does not provide absolute identification when all the used invariants cannot distinguish between two or more different knots. In that case, a list of possible candidates for $L$ is returned.