Make customized submit sort column sortable

My web site has two customized sorts: Public sale and Lot. Tons are linked to an Public sale by way of the post_parent area. I added an Public sale column to the Tons record within the admin space.

perform theme_lot_custom_columns($columns){
    $new = array();
    foreach($columns as $key => $worth) {
        if ($key=='date') {
            // Put the Public sale column earlier than the Date column
            $new['auction'] = __('Public sale');
        }
        $new[$key] = $worth;
    }
return $new;
}
add_filter('manage_lot_posts_columns', 'theme_lot_custom_columns');

perform theme_lot_custom_column($column, $post_id) {
    if ($column === 'public sale'){
        $submit = get_post($post_id);
        if ($post->post_parent) {
            $post_parent = get_post($post->post_parent);
            echo '<a href="' . get_edit_post_link($post_parent->ID). '">' . $post_parent->post_title . '</a>';
        }
    }
}
add_action('manage_lot_posts_custom_column', 'theme_lot_custom_column', 5, 2);

I would prefer to make this Public sale column sortable alphabetically by the public sale title. I believed that including the next code was sufficient, however no.

perform theme_lot_sortable_columns($columns) {
    $columns['auction'] = 'public sale';
    return $columns;
}   
add_filter('manage_edit-lot_sortable_columns', 'theme_lot_sortable_columns');

Any assist could be appreciated.

Thanks!