LaTeX symbol for the Kulkarni-Nomizu product

You can combine \bigcirc with \wedge.

Of course, you can also create and use a new command. The result is, obviously, the same with or without the new command.

Edit:

However, with \mspace the symbol becomes misaligned when used in \scriptstyle or - worse - in \scriptscriptstyle.

\documentclass{article}
\usepackage{amsmath, amssymb}

% Of course, you can create...
\newcommand{\KN}{\mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}}}

\begin{document}
\[
h \mathbin{\bigcirc\mspace{-15mu}\wedge\mspace{3mu}} k
\]
% ... and use a new command:
\[
h \KN k
\]
\[
x_{h \KN k}
\]
\[
y_{x_{h \KN k}}
\]
\end{document}

See the output:

A solution with \ooalign.

\documentclass{article}

\makeatletter
\newcommand*\owedge{\mathpalette\@owedge\relax}
\newcommand*\@owedge[1]{%
  \mathbin{%
    \ooalign{%
      $#1\m@th\bigcirc$\cr
      \hidewidth$#1\m@th\wedge$\hidewidth\cr
    }%
  }%
}
\makeatother

\begin{document}

$h \owedge k_{h \owedge k_{h \owedge k}}$

\end{document}