Link to the next/prev posts in index loop in same category?

Naughty boy you, you should never use query_posts. Straight from the codex:

Note: This function isn’t meant to be used by plugins or themes. As
explained later, there are better, more performant options to alter
the main query. query_posts() is overly simplistic and problematic way
to modify main query of a page by replacing it with new instance of
the query.

WP_Query should rather be used. I would do something like this

<?php
// set the "paged" parameter (use 'page' if the query is on a static front page)
$paged = ( get_query_var( 'paged' ) ) ? get_query_var( 'paged' ) : 1;

// the query
$the_query = new WP_Query( 'cat=56&paged=' . $paged ); 
?>

<?php if ( $the_query->have_posts() ) : ?>

<?php
// the loop
while ( $the_query->have_posts() ) : $the_query->the_post(); 
?>
<?php the_title(); ?>
<?php endwhile; ?>

<?php

// next_posts_link() usage with max_num_pages
next_posts_link( 'Older Entries', $the_query->max_num_pages );
previous_posts_link( 'Newer Entries' );
?>