Line intersection jump causes arrow to be drawn from the center of node instead of the boundary

Question

I have a diagram where the arrow lines between \nodes overlap and I would like them to have a semicircle (or some sort of indication really) on their intersection point to communicate which arrow is going where. I have been consulting this answer for my task. The 'flexible' solution is working as intended and, as aptly named, works with -|, |- variants of the connection which I have a need for.

However, the arrow lines seem to originate from the center of my nodes a and b like so. I would have hoped from them to look like that of the arrow between x and y just with the hump added.

My attempt.

\documentclass[10pt, border=1in]{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric, arrows, arrows.meta}
\usetikzlibrary{positioning}
\usetikzlibrary{calc}
\tikzstyle{object} = [draw, rectangle, minimum width=2cm, minimum height=1cm, text centered, draw=black]
\tikzstyle{arrow} = [thick,->,>=Triangle]
\pgfmathsetmacro{\jumpswap}{1}
\tikzset{
    % set up keys for radius, position, swap
    jump radius/.estore in=\jumpradius,
    jump pos/.estore in=\jumppos,
    jump swap/.code={\pgfmathsetmacro{\jumpswap}{\jumpswap-1}},
    jump radius=0.15cm,
    jump pos=0.5,
    % set up styles for the various to-paths
    -u-/.style={ % straight line
        to path={
            let \p1=(\tikztostart),\p2=(\tikztotarget),\n1={atan2(\y2-\y1,\x2-\x1)} in
            (\p1) -- ($($(\p1)!\jumppos!(\p2)$)!\jumpradius!(\p1)$)
            arc[start angle=\n1+180,delta angle=-180\jumpswap,radius=\jumpradius] -- (\p2)}
    },
    -u|/.style={ % -| path with jump on horizontal leg
        to path={
            let \p1=(\tikztostart),\p2=(\tikztostart-|\tikztotarget), \n1={atan2(\y2-\y1,\x2-\x1)} in
            (\p1) -- ($($(\p1)!\jumppos!(\p2)$)!\jumpradius!(\p1)$)
            arc[start angle=\n1+180,delta angle=-180\jumpswap,radius=\jumpradius] --(\p2) -- (\tikztotarget)}
    },
    |u-/.style={ % |- path with jump on vertical leg
        to path={
            let \p1=(\tikztostart),\p2=(\tikztostart|-\tikztotarget), \n1={atan2(\y2-\y1,\x2-\x1)} in
            (\p1) -- ($($(\p1)!\jumppos!(\p2)$)!\jumpradius!(\p1)$)
            arc[start angle=\n1+180,delta angle=-180\jumpswap,radius=\jumpradius] -- (\p2) -- (\tikztotarget)}
    },
    -|u/.style={ % -| path with jump on vertical leg
        to path={
            let \p1=(\tikztostart-|\tikztotarget),\p2=(\tikztotarget), \n1={atan2(\y2-\y1,\x2-\x1)} in
            (\tikztostart) -- (\p1) -- ($($(\p1)!\jumppos!(\p2)$)!\jumpradius!(\p1)$)
            arc[start angle=\n1+180,delta angle=-180\jumpswap,radius=\jumpradius] -- (\p2)}
    },
    |-u/.style={ % |- path with jump on horizontal leg
        to path={
            let \p1=(\tikztostart|-\tikztotarget),\p2=(\tikztotarget), \n1={atan2(\y2-\y1,\x2-\x1)} in
            (\tikztostart) -- (\p1) -- ($($(\p1)!\jumppos!(\p2)$)!\jumpradius!(\p1)$)
            arc[start angle=\n1+180,delta angle=-180\jumpswap,radius=\jumpradius] -- (\p2)}
    },
    % define the jump style, set it to use straight line by default
    jump/.style={-u-,#1},
    jump/.default={}
}
\begin{document}
\begin{tikzpicture}[node distance=2cm]
    \node(n1)[object]{x};
    \node(n2)[object, below of=n1, xshift=-3.5cm]{a};
    \node(n3)[object, right of=n2, xshift=5cm]{b};
    \node(n4)[object, below of=n1, yshift=-2cm]{y};

    \draw[arrow](n1)--(n4);
    \draw[arrow](n2) to[jump] (n3); % connecting arrow called here
\end{tikzpicture}


\end{document}

I am open to any other solution that enables displaying the desired intersection marker(s). Thank you for reading.

Answer 1

Here's an alternative method of achieving the jumping arc which uses my spath3 library. It locates the intersection of the paths and automatically places an arc at the intersection point in the over path. It additionally breaks the under path at the arc.

\documentclass[10pt, border=1in]{standalone}
%\url{https://tex.stackexchange.com/q/594585/86}
\usepackage{tikz}
\usetikzlibrary{
  shapes.geometric,
  arrows.meta,
  positioning,
  calc,
  intersections,
  spath3
}
\tikzset{
  object/.style={
    draw,
    rectangle,
    minimum width=2cm,
    minimum height=1cm,
    text centered,
    draw=black
  },
  arrow/.style={
    thick,
    ->,
    >=Triangle
  },
  bridging path/.initial=arc,
  bridging span/.initial=8pt,
  bridging gap/.initial=4pt,
  bridge/.style 2 args={
    spath/split at intersections with={#1}{#2},
    spath/insert gaps after
    components={#1}{\pgfkeysvalueof{/tikz/bridging span}},
    spath/join components upright
    with={#1}{\pgfkeysvalueof{/tikz/bridging path}},
    spath/split at intersections with={#2}{#1},
    spath/insert gaps after
    components={#2}{\pgfkeysvalueof{/tikz/bridging gap}},
  }
}
\AtBeginDocument{
  \tikz[overlay] \path[spath/save=arc] (0,0) arc[radius=1cm, start
    angle=180, delta angle=-180];
}
\begin{document}
\begin{tikzpicture}[node distance=2cm]
    \node(n1)[object]{x};
    \node(n2)[object, below of=n1, xshift=-3.5cm]{a};
    \node(n3)[object, right of=n2, xshift=5cm]{b};
    \node(n4)[object, below of=n1, yshift=-2cm]{y};


\path[spath/save=up] (n1) -- (n4);
\path[spath/save=along] (n2) -- (n3);
\tikzset{bridge={along}{up}}
\draw[arrow,spath/use=up];
\draw[arrow,spath/use=along];
\end{tikzpicture}


\end{document}