RegionPlot3D KnotData

You can feed the BoundaryMeshRegion[] you can obtain from KnotData[] into RegionPlot3D[]. For example:

trefBMR = KnotData["Trefoil", "BoundaryMeshRegion"];
RegionPlot3D[RegionMember[trefBMR, {x, y, z}],
             {x, -7/2, 7/2}, {y, -7/2, 7/2}, {z, -3/2, 3/2},
             BoxRatios -> Automatic, Lighting -> "Neutral",
             Mesh -> None, PlotPoints -> 35, 
             PlotStyle -> Directive[Texture[ExampleData[{"ColorTexture",
                                                         "WhiteMarble"}]]]]

Edit

n = 15;
vor = VoronoiMesh[
   RandomPoint[Rectangle[{0, 0}, {2 π, 2 π}], 
    n], {{0, 2 π}, {0, 2 π}}];
polys = MeshPrimitives[vor, 2];
g = Show[Table[{Red, 
      Disk[x /. Last[#], Abs@First[#]] &@
       NMinimize[SignedRegionDistance[poly][x], 
        x ∈ poly]}, {poly, polys}] // Graphics];
curve3 = KnotData["Trefoil", "SpaceCurve"];
basis = Last[FrenetSerretSystem[curve3[t], t]];
{tangent, normal, binormal} = basis;
ParametricPlot3D[
 curve3[t] + .6 (Cos[u]*normal + Sin[u]*binormal), {u, 0, 
  2 π}, {t, 0, 2 π}, PlotPoints -> 80, Mesh -> None, 
 Boxed -> False, Axes -> False, PlotStyle -> Texture[g], 
 TextureCoordinateScaling -> True, 
 TextureCoordinateFunction -> Function[{x, y, z, u, t}, {u, 9 t}], 
 ViewPoint -> {0.2, -0.3, 3.3}]

Original

A starting point.

curve3 = KnotData["Trefoil", "SpaceCurve"];
basis = Last[FrenetSerretSystem[curve3[t], t]];
{tangent, normal, binormal} = basis;
g = Graphics[{Red, Disk[{0, 0}, .5]}, PlotRangePadding -> .5];
ParametricPlot3D[
 curve3[t] + .6 (Cos[u]*normal + Sin[u]*binormal), {u, 0, 
  2 π}, {t, 0, 2 π}, PlotPoints -> 80, Mesh -> None, 
 Boxed -> False, Axes -> False, PlotStyle -> Texture[g], 
 TextureCoordinateScaling -> False, 
 TextureCoordinateFunction -> Function[{x, y, z, t, u}, {x, y}], 
 ViewPoint -> {0.2, -0.3, 3.3}]

SliceContourPlot3D[Sin[5 x] Sin[6 y] Sin[4 z], 
 KnotData["Trefoil", "Region"],
 {x, -Pi, Pi}, {y, -Pi, Pi}, {z, -Pi, Pi}, 
 Contours -> {-1/6, 1/6}, ContourStyle -> None, 
 ContourShading -> {White, Red}, 
 Boxed -> False, ImageSize -> Large, Axes -> False, PlotPoints -> 90]