Tikz : self intersection not recognized in knots library of spath3

As you say, it is the end tolerance parameter that is at stake. But, I redefined your path because its departure and arrival use uselessly beziers curves to draw straight lines. This way I get the result you want.

\documentclass[tikz,border=5mm]{standalone}    
\usetikzlibrary{calc,knots}    

%set default length and width parameters
\def\x{.7}
\def\y{1}

\begin{document}
 \begin{tikzpicture}
  \begin{knot}[consider self intersections,end tolerance=.01pt,
  %draft mode=crossings
  ]
   \strand (0,-\y) %.. controls +(0,0) <-- useless beziers curve
      %.. 
      --(0,0) .. controls +(0,\x/3) and +(0,\x/3) 
      .. (\x/2,0) .. controls +(0,-\x/3) and +(0,-\x/3) 
      .. (0,0) %.. controls +(0,0) .. <- useless beziers curve
       --(0,\y);
  \end{knot}
 \end{tikzpicture}
\end{document}

LoopSpace explained very well what the issue is, and how to fix it. Here I'd like to draw your attention to an alternative, which has been used in a very similiar application. This alternative is by no means "better" than, or even comparable to, the knots package. The reason why I am mentioning it here is that it gives a rather reasonable output without the need modify your path, or to tune parameters.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{decorations.pathreplacing}
\begin{document}
\begin{tikzpicture}[rubout/.style={/utils/exec=\tikzset{rubout/.cd,#1},
 decoration={show path construction,
      curveto code={
       \draw [white,line width=\pgfkeysvalueof{/tikz/rubout/line width}+2*\pgfkeysvalueof{/tikz/rubout/halo}] 
        (\tikzinputsegmentfirst) .. controls
        (\tikzinputsegmentsupporta) and (\tikzinputsegmentsupportb)  ..(\tikzinputsegmentlast); 
       \draw [line width=\pgfkeysvalueof{/tikz/rubout/line width},shorten <=-0.1pt,shorten >=-0.1pt] (\tikzinputsegmentfirst) .. controls
        (\tikzinputsegmentsupporta) and (\tikzinputsegmentsupportb) ..(\tikzinputsegmentlast);  
      }}},rubout/.cd,line width/.initial=0.6pt,halo/.initial=0.6pt]
  \def\x{.7}
  \def\y{1}
  \draw[rubout,decorate] (0,-\y) .. controls +(0,0) 
      .. (0,0) .. controls +(0,\x/3) and +(0,\x/3) 
      .. (\x/2,0) .. controls +(0,-\x/3) and +(0,-\x/3) 
      .. (0,0) .. controls +(0,0) .. (0,\y);\end{tikzpicture}
\end{document}

Thanks for all the answers !
Following the suggestion of Loop Space I created a more stable variant by slightly deviating from the initial picture. I also realized that, for this picture, using arc instead of beziers curves keeps the code more readable, but that's just taste I guess.

\documentclass[tikz, border=5mm]{standalone}
\usetikzlibrary{calc,knots}
% basic distances for tikz
\def\y{1}
\def\r{-.2}

\begin{document}        
  \begin{tikzpicture}
    \begin{knot}[consider self intersections, end tolerance=1pt]
      \strand (0,-\y) to (0,0) arc (0:90:\r) arc (90:270:\r*.6) arc (270:360:\r)  
                      to (0,\y);
    \end{knot}
  \end{tikzpicture}  
\end{document}