Pensieve Header: A little on Gauss Diagrams.

<< KnotTheory`
GD[K_] := GD @@ (
PD[K] /. X[i_,j_,k_,l_] :> If[PositiveQ[X[i,j,k,l]],
Ar[l,i,+1], Ar[j,i,-1]
]
)

GD[Knot[8,11]]

Draw[expr_] := expr /. gd_GD:> Draw[gd];
Draw[gd_GD] := Module[
{n=Length[gd], h, k=0},
Graphics[{
Line[{{0,0}, {2n+1,0}}],
Table[Text[i, {i,-0.3}], {i,2n}],
(List@@gd) /.  {
Ar[i_, j_, s_] :> {
h=Abs[i-j]/2;
BezierCurve[{
{i,0},{i,h}, {(i+j)/2, h}, {j,h}, {j,0}
}, SplineDegree -> 2],
Text[s*(++k), {(i+j)/2, h-0.3}],
Line[{{j-0.2, 0.4}, {j,0}, {j+0.2, 0.4}}]
}
}
}
]
];
Draw[GD[Knot[8,11]]]

Draw[GD[#]]& /@ AllKnots[7]

Draw[GD[#]]& /@ AllKnots[8]