Shade a region in a knot diagram

Question

Using the knots package, I have drawn a (3,7)-torus knot. I would like for the inner region of the diagram to be shaded gray.

A minimal working example for the code follows.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{hobby,knots}
\begin{document}

\begin{tikzpicture}[use Hobby shortcut]
\fill[black!30!white] (90:1) \foreach \k in {1,...,7}{--(90+\k*360/7:1)};

\begin{knot}[
  consider self intersections=true,
  ignore endpoint intersections = false,
  flip crossing/.list={1,7,6,11,10, 13, 15, 2}
]
\strand[very thick] ([closed]90:2) foreach \k in {1,...,7} 
   { .. (90-360/7+\k*1080/7:1) .. (90+\k*1080/7:2) } (90:2);
\end{knot}

\end{tikzpicture}
\end{document}

The figure produced by this code follows.

The problem with the figure is that I would like the gray region to fit neatly next to the arcs of the knot diagram that bounds it. There are three issues I would like to fix.

1. The knot tikzlibrary works by drawing larger white background lines around the strands that appear on top at a crossing. This is how the effect of an over/under crossing is achieved (I think). I want the shaded region to ignore those larger white lines and meet the actual arcs of the knot diagram.
2. I've guessed the crossing coordinates in my \fill command. Is there anyway to compute exactly what those coordinates are?
3. The arcs in the inner region are closely approximated by straight line segments. In this case, I could probably get away by filling a path (if question 1 was resolved). However, I do not know how to approach shading some of the regions with curvier boundaries.

Any help is greatly appreciated.

Answer 1

I have no idea how to fill exactly this area, but you can cheat like this :

\documentclass[tikz,border=7pt]{standalone}
\usetikzlibrary{hobby,knots,shapes}

\begin{document}

  \begin{tikzpicture}[use Hobby shortcut]

    \begin{knot}[
        consider self intersections=true,
        ignore endpoint intersections = false,
        flip crossing/.list={1,7,6,11,10, 13, 15, 2}
      ]
      \strand[very thick] ([closed]90:2) foreach \k in {1,...,7}
        { .. (90-360/7+\k*1080/7:1) .. (90+\k*1080/7:2) } (90:2);
    \end{knot}

    \node[regular polygon,regular polygon sides=7,fill,fill opacity=.21, inner sep=6.8mm, rotate=7] {};
  \end{tikzpicture}
\end{document}

Answer 2

Here's a method using blend modes (the PGF manual warns that these might not be supported by all renderers/printers). The thinking behind it is that the white-out part of the knot should blank out other strands of the knot but not anything else. So everything ends up in a blend group and we choose the blend modes appropriately.

To get the actual region, we fill the knot path using the even odd rule and clip against a circle of just the right size (magic numbers courtesy of www.i-just-made-them-up.com). This gets us the correct region (for a more generic shape, one might have to be a bit creative for this part). Drawing the knot over the top sort of works, but has the issue of the white-out showing and hence the use of blend modes.

\documentclass{article}
%\url{https://tex.stackexchange.com/q/410096/86}
\usepackage{tikz}
\usetikzlibrary{hobby,knots}

\begin{document}

\begin{tikzpicture}[use Hobby shortcut]

\begin{scope}[blend group=darken]

\begin{scope}[blend group=normal,even odd rule]
\clip (0,0) circle[radius=1.07];
\fill[black!30!white, save Hobby path={knot}] ([closed]90:2) foreach \k in {1,...,7} 
   { .. (90-360/7+\k*1080/7:1) .. (90+\k*1080/7:2) } (90:2);
\end{scope}

\begin{scope}[blend group=normal]
\begin{knot}[
  consider self intersections=true,
  ignore endpoint intersections = false,
  flip crossing/.list={1,7,6,11,10, 13, 15, 2}
]
   \strand[very thick] ([closed]90:2) foreach \k in {1,...,7} { .. (90-360/7+\k*1080/7:1) .. (90+\k*1080/7:2) } (90:2);
\end{knot}
\end{scope}

\end{scope}
\end{tikzpicture}
\end{document}

Result:

Detail of a single junction: