3

I have realised that my knot diagram changes one of its crossings when it's scaled differently. The crossings should be correct in the second diagram generated by the code below. I don't know how this has happened. I have tried using draft mode=crossings but it was no use. Could I please have some help in switching the crossings to the correct version in the first diagram?

This is my code:

\documentclass[a4paper,11pt]{report}
\usepackage{graphicx,amssymb,amstext,amsmath}
\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing,decorations.markings,hobby,knots,celtic,shapes.geometric,calc}

\begin{document}

\tikzset{
  knot diagram/every strand/.append style={
    line width=1.5pt,violet
  },
}

\tikzset{
  my style/.style={
    dashed
  }
}

\begin{tikzpicture}[scale=0.9, use Hobby shortcut, add arrow/.style={postaction={decorate}, decoration={
  markings,
  mark=at position 0.25 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.653 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.75 with {\arrow[line width=1.5pt]{<}}}}]
\begin{knot}[
  consider self intersections=true,
  ignore endpoint intersections=false,
  flip crossing=3,
  rotate=180]
  \strand ([closed]0,0) .. (0.7,-0.7) .. (0,-1) .. (-1.7,0) .. (-1.5,2.2) .. (0,2.5) .. (.5,2) .. (-.5,1) [add arrow,violet] .. (.5,0) .. (0,-.5) .. (-.5,0) .. (.5,1) .. (-.5,2) .. (-1.5,1) .. (0,0);
  \strand[my style, lightgray] (-0.7,-1.2) -- (-0.7,-.9); 
  \strand[my style, lightgray] (-0.7,-.75) -- (-0.7,2.5);
\end{knot}
\path (0,-.7);
\end{tikzpicture}

%---------------------------------------------------------------

\begin{tikzpicture}[scale=2, use Hobby shortcut, add arrow/.style={postaction={decorate}, decoration={
  markings,
  mark=at position 0.25 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.653 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.75 with {\arrow[line width=1.5pt]{<}}}}]
\begin{knot}[
  consider self intersections=true,
  ignore endpoint intersections=false,
  flip crossing=3,
  rotate=180]
  \strand ([closed]0,0) .. (0.7,-0.7) .. (0,-1) .. (-1.7,0) .. (-1.5,2.2) .. (0,2.5) .. (.5,2) .. (-.5,1) [add arrow,violet] .. (.5,0) .. (0,-.5) .. (-.5,0) .. (.5,1) .. (-.5,2) .. (-1.5,1) .. (0,0);
  \strand[my style, lightgray] (-0.7,-1.2) -- (-0.7,-.9); 
  \strand[my style, lightgray] (-0.7,-.75) -- (-0.7,2.5);
\end{knot}
\path (0,-.7);
\end{tikzpicture}

\end{document}
2
  • The problem is the presence of the dashed line. Do you need that to be defined as part of the knot? Could it be an ordinary line drawn before the knot is rendered? I'll post some different possibilities when I get back to a proper computer. Apr 3, 2019 at 23:27
  • This is the same underlying problem as addressed in tex.stackexchange.com/q/323758/86 I'll post an answer here as well as it's not easy to generalise, but refer to the answer there for the explanation. Apr 4, 2019 at 16:35

2 Answers 2

2

The underlying issue here is the same as noted in Unexpected behavior of the "knots" tikz library when a crossing is close to another (thick) line: What's going on?. There are two crossings that are very close together and the clipping circles (when the crossings are drawn) overlap, meaning that when the violet line is drawn over the dashed grey line, it also draws over itself at that problematic crossing.

Here are a few possible fixes. In the first, I reduce the radius of the clipping circles. In the second, I took the dashed grey line out of the knot as it doesn't seem to be actually part of the knot (note that if you do want to keep it in the knot, you should use the key only when rendering for the dashed part - see the documentation for details). In the third fix, I simplified your hobby path a little.

Results:

The various fixes

Code:

\documentclass[a4paper,11pt]{report}
%\url{https://tex.stackexchange.com/q/478294/86}
\usepackage{shellesc}
\usepackage{graphicx,amssymb,amstext,amsmath}
\usepackage{tikz}
\usetikzlibrary{
  decorations.pathreplacing,
  decorations.markings,
  hobby,
  knots,
  celtic,
  shapes.geometric,
  calc,
  external
}

\tikzexternalize

\begin{document}

\tikzset{
  knot diagram/every strand/.append style={
    line width=1.5pt,violet
  },
}

\tikzset{
  my style/.style={
    dashed
  }
}




Demonstration of the problem: the clipping circles are big enough that the circle associated with the crossing between the violet line and the dashed grey line overlaps the problematic crossing.

\tikzsetnextfilename{problem}
\begin{tikzpicture}[scale=0.9, use Hobby shortcut, add arrow/.style={postaction={decorate}, decoration={
  markings,
  mark=at position 0.25 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.653 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.75 with {\arrow[line width=1.5pt]{<}}}}]
\begin{knot}[
  consider self intersections=true,
  ignore endpoint intersections=false,
  flip crossing=3,
  background clip/.append style={
    preaction={
      fill=gray,
      fill opacity=.5,
    }
  },
  rotate=180]
  \strand ([closed]0,0) .. (0.7,-0.7) .. (0,-1) .. (-1.7,0) .. (-1.5,2.2) .. (0,2.5) .. (.5,2) .. (-.5,1) [add arrow,violet] .. (.5,0) .. (0,-.5) .. (-.5,0) .. (.5,1) .. (-.5,2) .. (-1.5,1) .. (0,0);
  \strand[knot diagram/only when rendering/.style={my style}, lightgray] (-0.7,-1.2) -- (-0.7,-.9); 
  \strand[knot diagram/only when rendering/.style={my style}, lightgray] (-0.7,-.75) -- (-0.7,2.5);
\end{knot}
\path (0,-.7);
\end{tikzpicture}

%---------------------------------------------------------------

Fix one: shrink the clipping circles using the \verb+clip radius+ key.

\tikzsetnextfilename{fix-one}
\begin{tikzpicture}[scale=0.9, use Hobby shortcut, add arrow/.style={postaction={decorate}, decoration={
  markings,
  mark=at position 0.25 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.653 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.75 with {\arrow[line width=1.5pt]{<}}}}]
\begin{knot}[
  consider self intersections=true,
  ignore endpoint intersections=false,
  flip crossing=3,
  clip radius=3pt,
  background clip/.append style={
    preaction={
      fill=gray,
      fill opacity=.5,
    }
  },
  rotate=180]
  \strand ([closed]0,0) .. (0.7,-0.7) .. (0,-1) .. (-1.7,0) .. (-1.5,2.2) .. (0,2.5) .. (.5,2) .. (-.5,1) [add arrow,violet] .. (.5,0) .. (0,-.5) .. (-.5,0) .. (.5,1) .. (-.5,2) .. (-1.5,1) .. (0,0);
  \strand[knot diagram/only when rendering/.style={my style}, lightgray] (-0.7,-1.2) -- (-0.7,-.9); 
  \strand[knot diagram/only when rendering/.style={my style}, lightgray] (-0.7,-.75) -- (-0.7,2.5);
\end{knot}
\path (0,-.7);
\end{tikzpicture}

Fix two: the dashed grey lines don't appear to actually be anything to do with the knot, so draw them separately.
This means that there aren't two crossings near to each other any more.

\tikzsetnextfilename{fix-two}
\begin{tikzpicture}[scale=0.9, use Hobby shortcut, add arrow/.style={postaction={decorate}, decoration={
  markings,
  mark=at position 0.25 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.653 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.75 with {\arrow[line width=1.5pt]{<}}}}]
\draw[
  line width=1.5pt,
  my style,
  lightgray,
  rotate=180
] (-0.7,-1.2) -- (-0.7,2.5);
\begin{knot}[
  consider self intersections=true,
  ignore endpoint intersections=false,
%  flip crossing=6,
  rotate=180]
  \strand ([closed]0,0) .. (0.7,-0.7) .. (0,-1) .. (-1.7,0) .. (-1.5,2.2) .. (0,2.5) .. (.5,2) .. (-.5,1) [add arrow,violet] .. (.5,0) .. (0,-.5) .. (-.5,0) .. (.5,1) .. (-.5,2) .. (-1.5,1) .. (0,0);
\end{knot}
\path (0,-.7);
\end{tikzpicture}

Fix three: you specify a lot of \emph{nodes} in your Hobby path.
If you aren't wedded to that specific path then reducing the number of nodes can simplify the knot algorithm.

\tikzsetnextfilename{fix-three}
\begin{tikzpicture}[scale=0.9, use Hobby shortcut, add arrow/.style={postaction={decorate}, decoration={
  markings,
  mark=at position 0.2 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.28 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.8 with {\arrow[line width=1.5pt]{<}}}}]
\draw[
  line width=1.5pt,
  my style,
  lightgray,
  rotate=180
] (-0.7,-2.2) -- (-0.7,3.5);
\begin{knot}[
  consider self intersections=true,
  end tolerance=3pt,
  flip crossing=4,
  rotate=180]
  \strand ([closed].5,2) [add arrow,violet] ..  (-.5,1) .. (.3,-.5) .. (-.3,-.5) .. (.5,1) .. ([tension=3]-1.5,1) .. (.7,-1) .. (-2,1);
\end{knot}
\path (0,-.7);
\end{tikzpicture}


\tikzsetnextfilename{large-scale}
\begin{tikzpicture}[scale=2, use Hobby shortcut, add arrow/.style={postaction={decorate}, decoration={
  markings,
  mark=at position 0.25 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.653 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.75 with {\arrow[line width=1.5pt]{<}}}}]
\begin{knot}[
  consider self intersections=true,
  ignore endpoint intersections=false,
  flip crossing=3,
  rotate=180]
  \strand ([closed]0,0) .. (0.7,-0.7) .. (0,-1) .. (-1.7,0) .. (-1.5,2.2) .. (0,2.5) .. (.5,2) .. (-.5,1) [add arrow,violet] .. (.5,0) .. (0,-.5) .. (-.5,0) .. (.5,1) .. (-.5,2) .. (-1.5,1) .. (0,0);
  \strand[knot diagram/only when rendering/.style={my style}, lightgray] (-0.7,-1.2) -- (-0.7,-.9); 
  \strand[knot diagram/only when rendering/.style={my style}, lightgray] (-0.7,-.75) -- (-0.7,2.5);
\end{knot}
\path (0,-.7);
\end{tikzpicture}

\end{document}
2

I did precisely what you suggested, draft mode=crossings, after which I flipped crossing number 6 and get

\documentclass[a4paper,11pt]{report}
\usepackage{graphicx,amssymb,amstext,amsmath}
\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing,decorations.markings,hobby,knots,celtic,shapes.geometric,calc}

\begin{document}

\tikzset{
  knot diagram/every strand/.append style={
    line width=1.5pt,violet
  },
}

\tikzset{
  my style/.style={
    dashed
  }
}

\begin{tikzpicture}[scale=0.9, use Hobby shortcut, add arrow/.style={postaction={decorate}, decoration={
  markings,
  mark=at position 0.25 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.653 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.75 with {\arrow[line width=1.5pt]{<}}}}]
\begin{knot}[
  consider self intersections=true,
  ignore endpoint intersections=false,
  flip crossing=3,
  rotate=180]
  \strand ([closed]0,0) .. (0.7,-0.7) .. (0,-1) .. (-1.7,0) .. (-1.5,2.2) .. (0,2.5) .. (.5,2) .. (-.5,1) [add arrow,violet] .. (.5,0) .. (0,-.5) .. (-.5,0) .. (.5,1) .. (-.5,2) .. (-1.5,1) .. (0,0);
  \strand[my style, lightgray] (-0.7,-1.2) -- (-0.7,-.9); 
  \strand[my style, lightgray] (-0.7,-.75) -- (-0.7,2.5);
\end{knot}
\path (0,-.7);
\end{tikzpicture}

%---------------------------------------------------------------

\begin{tikzpicture}[scale=2, use Hobby shortcut, add arrow/.style={postaction={decorate}, decoration={
  markings,
  mark=at position 0.25 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.653 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.75 with {\arrow[line width=1.5pt]{<}}}}]
\begin{knot}[%draft mode=crossings,
  consider self intersections=true,
  ignore endpoint intersections=false,
  flip crossing/.list={3,6},
  rotate=180]
  \strand ([closed]0,0) .. (0.7,-0.7) .. (0,-1) .. (-1.7,0) .. (-1.5,2.2) .. (0,2.5) .. (.5,2) .. (-.5,1) [add arrow,violet] .. (.5,0) .. (0,-.5) .. (-.5,0) .. (.5,1) .. (-.5,2) .. (-1.5,1) .. (0,0);
  \strand[my style, lightgray] (-0.7,-1.2) -- (-0.7,-.9); 
  \strand[my style, lightgray] (-0.7,-.75) -- (-0.7,2.5);
\end{knot}
\path (0,-.7);
\end{tikzpicture}

\end{document}

enter image description here

OK, I am not able to solve the problem. Here is a proposal for a workaround.

\documentclass[a4paper,11pt]{report}
\usepackage{graphicx,amssymb,amstext,amsmath}
\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing,decorations.markings,hobby,knots,celtic,shapes.geometric,calc}

\begin{document}

\tikzset{
  knot diagram/every strand/.append style={
    line width=3.2pt,violet
  },
}

\tikzset{
  my style/.style={
    dashed
  }
}

\scalebox{0.45}{
\begin{tikzpicture}[scale=2, use Hobby shortcut, add arrow/.style={postaction={decorate}, decoration={
  markings,
  mark=at position 0.25 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.653 with {\arrow[line width=1.5pt]{<}},
  mark=at position 0.75 with {\arrow[line width=1.5pt]{<}}}}]
\begin{knot}[%draft mode=crossings,
  consider self intersections=true,
  ignore endpoint intersections=false,
  flip crossing=3,
  rotate=180]
  \strand ([closed]0,0) .. (0.7,-0.7) .. (0,-1) .. (-1.7,0) .. (-1.5,2.2) .. (0,2.5) .. (.5,2) .. (-.5,1) [add arrow,violet] .. (.5,0) .. (0,-.5) .. (-.5,0) .. (.5,1) .. (-.5,2) .. (-1.5,1) .. (0,0);
  \strand[my style, lightgray] (-0.7,-1.2) -- (-0.7,-.9); 
  \strand[my style, lightgray] (-0.7,-.75) -- (-0.7,2.5);
\end{knot}
\path (0,-.7);
\end{tikzpicture}}

\end{document}

enter image description here

5
  • 1
    Ah thank you (as always!) - but I meant how could I change that crossing in the first diagram?
    – JpW
    Mar 7, 2019 at 21:42
  • @JpW Oh, sorry. Yes, I confirm that this is difficult. One can get the intersection as desired by putting flip crossing={8}, in the first graph, but then the intersection with the vertical line is wrong (I also removed (0,0) in the end). I suspect that the fact that you use the coordinate (0,0) in the end (in addition to [closed]) could be part of the reason.
    – user121799
    Mar 7, 2019 at 22:40
  • If you alter the scale from 1 to 2 without any changes to the crossing flips, then you can actually see the crossing slowly change...
    – JpW
    Mar 7, 2019 at 22:49
  • @JpW Yes. I see how there can be problems. TikZ uses approximations to find intersections. Sometimes it finds more than there actually are, i.e. it may overcount intersections and so on. This does depend on the scale. What you can always do is to put something that works in a scalebox (and scale the line width and the nodes up with transform shape). Is your aim to have the second plot scaled down to the first one?
    – user121799
    Mar 7, 2019 at 22:54
  • Yes it is. I'm writing up my report, which is on a much smaller scale than my poster (which was what I previously used). I wrote the poster in beamer several weeks ago and I'm trying to use the same code for the report but that crossing is a headache.
    – JpW
    Mar 7, 2019 at 22:59

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