Tikz Braids: How to draw Singular Braids (intersections)?

Question

I've been using the TikZ braids package, which works great. Unfortunately, I now need to start drawing singular braids/knots, which I don't think the package handles. These concepts are "less popular" than regular braids/knots so it is understandable that it hasn't been developed.

Here's a basic example I pulled off. The first crossing, illustrated with a black dot, is the singular crossing. These are typically denoted with a black dot. The other crossings are completely normal.

My attempt:

My method to draw the singular crossing isn't great because

1. I had to manually draw a black circle
2. I had to guess the intersection position until I got it correctly

In short, this method is sort of a time-waster; but I'm going to have to draw lots of these...

My questions: Does anyone have any ideas or suggestions on a more systematic way of doing this? Perhaps a way to access a list of intersections (which I assume the package computes behind the scenes) which I could use to my advantage? I try to look for ways to avoid wasting time before I give up and accept wasting time.

Ideally, the package could be extended so that whenever a user types t_1, for example, instead of s_1, it detects that the user would like a singular crossing, so that it does all that guesswork/drawing for me. But obviously I'm completely incapable of doing that myself.

My code:

\documentclass[12pt,letterpaper]{article}
\usepackage{tikz} 
\usetikzlibrary{braids} 

\begin{document}
\begin{center}
    \def\height{3.5} %bar lengths
    \def\width{3.2} %bar difference heights, .2 added to make it nicer
    \begin{tikzpicture}
        \pic[braid/.cd, 
        number of strands = 3,
        thick]
        {braid={ s_1, s_2[braid/gap=0], s_3}};
        \filldraw (0.5,-0.75) circle (0.07cm); %the singular intersection
        \draw[thick] (-0.2,0) -- (\width,0);   %top bar
        \draw[thick] (-0.2,-\height) -- (\width,-\height); %bottom bar
        \end{tikzpicture}

        \emph{$\tau_1\sigma_2\sigma_3$}
\end{center}

\end{document}  

Answer 1

The knots package has more advanced options but a very blunt way to marking the crossing is to recover it via intersections of lines connecting the marks on the strands. Note also that, by using a local bounding box, you can avoid the hard coded dimensions for the horizontal bars.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{braids,calc}
\begin{document}
\begin{tikzpicture}
  \pic[local bounding box=my braid,braid/.cd, 
    number of strands = 3,
    thick,
    name prefix=braid]
    {braid={ s_1, s_2, s_3}};
    \filldraw (intersection of braid-2-0--braid-2-1 and braid-1-0--braid-1-1) 
     circle[radius=0.07cm];
    \draw[thick] ([xshift=-1ex]my braid.north west) --  ([xshift=1ex]my braid.north east)
        ([xshift=-1ex]my braid.south west) --  ([xshift=1ex]my braid.south east);
\end{tikzpicture}
\end{document}