[jQuery] Trouble getting parent anchor tag of child list

[dpat]

I have nested lists that are are trigged by clicking on an anchor
tag.  I hardcode a specific sublist to display initially, which
works.  The problem is parent anchors that have sublists get two
arrows, down and right, depending on if it's expanded or not, and I
can't get the down arrow to display initially.

I am trying to add a class to the parent anchor tag based on the
selected child sublist to expand.

function initMenu() {
                $('ul.mainmenu ul').hide();
                var expandDefaultMenu = $('ul.mainmenu ul:eq(0)'); //First 
submenu
expanded
                expandDefaultMenu.parent('a').addClass('defaultopen'); // <-- 
Cannot
get this to work.
                expandDefaultMenu.show();

                $('ul.mainmenu li a').click(
                        function() {
                                var checkElement = $(this).next();
                                if((checkElement.is('ul')) && 
(checkElement.is(':visible'))) {
                                        
$(this).css('background-image','url(/images/templates2008/
arrowright.gif)');
                                        checkElement.slideUp('normal');
                                        return false;
                                }
                                if((checkElement.is('ul')) && 
(!checkElement.is(':visible'))) {
                                        
$(this).css('background-image','url(/images/templates2008/
arrowdown.gif)');
                                        $('ul.mainmenu 
ul:visible').slideUp('normal');
                                        checkElement.slideDown('normal');
                                        return false;
                                }
                        }
                );
        }
        $(document).ready(function() { initMenu(); });