Visualize the TNB Basis of a Knot 

FrenetSerretSystem will compute the orthonormal basis associated to a curve in any dimension. In three dimensions, this basis is usually called the Tangent/Normal/Binormal or TNB basis.

Look up the function for the knot.

In[1]:=

X knot = KnotData["Trefoil", "SpaceCurve"];

Compute the associated TNB basis.

In[2]:=

X basis = Last[FrenetSerretSystem[knot[t], t]] // Simplify; {tangent, normal, binormal} = Map[Arrow[{knot[t], knot[t] + #}] &, basis];

Visualize the curve and its basis.

In[3]:=

X Manipulate[ Show[ParametricPlot3D[knot[s], {s, 0, 2 Pi}, PlotStyle -> Thick], Graphics3D[{Thick, Blue, tangent, Red, normal, Purple, binormal}], PlotRange -> 3, ImageSize -> Medium] // Evaluate, {t, 0, 2 Pi, Appearance -> {"Open", "Labelled"}}]

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