Link Categories in Main Template

I’m having trouble getting my links to show up in the main entry section of my template (as opposed to the sidebar.) I have a links page where I want all the links to show up by category, but when I put the php snippet in there nothing happens. What am I doing wrong?

Here is the page I’m talking about

And here are the codes I’ve tried:

<?php wp_get_linksbyname('Category Name') ?>
<?php get_links_list('id'); ?>
<?php wp_get_links(id); ?>
<?php get_links(ID, '<span>', '</span>', '', TRUE, 'url', FALSE); ?>