How do I create a major arc open to the left?

Here is a solution that uses \ooalign to overlay a white rectangle on part of a \medcirc.

\mathchoice is used to allow your symbol to appear in script or scriptscript styles. You can edit the \rule arguments to adjust height and thickness of the opening in each style.

\documentclass{amsart}
\usepackage{txfonts,color}

\newcommand{\arcl}{\mathchoice%
{\mathbin{\ooalign{\hfil$\medcirc$\hfil\cr\textcolor{white}{\rule[.25ex]{.2em}{.65ex}}\hfil\cr}}}
{\mathbin{\ooalign{\hfil$\medcirc$\hfil\cr\textcolor{white}{\rule[.25ex]{.2em}{.65ex}}\hfil\cr}}}
{\mathbin{\ooalign{\hfil$\scriptstyle\medcirc$\hfil\cr\textcolor{white}{\rule[.2ex]{.2em}{.4ex}}\hfil\cr}}}
{\mathbin{\ooalign{\hfil$\scriptscriptstyle\medcirc$\hfil\cr\textcolor{white}{\rule[.15ex]{.2em}{.25ex}}\hfil\cr}}}
}

\begin{document}

$a\medcirc b_{a\medcirc b_{a\medcirc b}}$

$a\arcl b_{a\arcl b_{a\arcl b}}$

\end{document}