Top level parent taxonomy ID

At the top of a product archive page I am trying to display a list of product categories. The rule should be that the list always shows the direct children of the top level.

There are several answers on WPA which I have been trying to implement (with my minimal skills) but I am missing something.

Here’s the query:

        <?php  
    $taxonomy     = 'product_cat';
    $orderby      = 'name';
    $order        = 'DESC';
    $show_count   = 0;      // 1 for yes, 0 for no
    $pad_counts   = 0;      // 1 for yes, 0 for no
    $hierarchical = 1;      // 1 for yes, 0 for no
    $title        = '';
    $empty        = 0;

   // So now I need to get the top parent id in a variable. I tried these 4 options

     /*
     // option 1
     $queried_object = get_queried_object();
    $term_id = $queried_object->term_id;


    // option 2
    $thisCat = get_category( get_query_var( 'cat' ) ); 
    while ( intval($thisCat->parent) > 0 ) {
      $thisCat = get_category ( $thisCat->parent );
     }
     $term_id = $thisCat->term_id

    //option 3
    $parent  = get_term_by( 'id', $term_id, $taxonomy);
    while ($parent->parent != 0){
        $parent  = get_term_by( 'id', $parent->parent, $taxonomy);
    }


    //option 4
    while ($catid) {
      $cat = get_category($catid); // get the object for the catid
      $catid = $cat->category_parent; // assign parent ID (if exists) to $catid
      // the while loop will continue whilst there is a $catid
      // when there is no longer a parent $catid will be NULL so we can assign our $catParent
      $catParent = $cat->cat_ID;
     }
    */

            // Testing if I have the correct value
    echo 'The returned variable is:'; print_r($term_id);



    $args2 = array(
      'taxonomy'     => $taxonomy,
      'child_of'     => $term_id,
      'orderby'      => $orderby,
      'order'        => $order,
      'show_count'   => $show_count,
      'pad_counts'   => $pad_counts,
      'hierarchical' => $hierarchical,
      'title_li'     => $title,
      'hide_empty'   => $empty
    );

    $sub_cats = get_categories( $args2 );

    $args = array(
        //'type'          => 'post',
        'orderby'       => 'term_group',
        'hide_empty'    => 0,
        'hierarchical'  => 0,
        'parent'        => 0,
        'taxonomy'      => 'product_cat'
    );
    $test = get_categories( $args ); 
    //print_r($test);

    $cat_id = get_query_var('cat');
    echo $cat_id;

    if (!empty($sub_cats)) { 

    echo '<ul class="categories">'; 

    foreach($sub_cats as $sub_category) {

        //echo 'de variabele sub_category'; print_r($sub_category);

        if ($sub_cats->$sub_category == 0) {
            $thumbnail_id  = get_woocommerce_term_meta( $sub_category->term_id, 'thumbnail_id', true );
            $image = wp_get_attachment_url( $thumbnail_id );
            $name = $sub_category->name;
            $slug = $sub_category->slug;

            echo '<li class="cat-item '.$slug.'"><a href="'. get_term_link($sub_category->slug, 'product_cat') .'"><img src="'.$image.'" " /><span class="txt">'.$name.'</span></a></li>';

            }
        }
        echo '<li class="cat-item search">';
        get_product_search_form();
        echo '</li>';

    echo '</ul>';

     } //endif !empty ?>

Thanks!
Juri