...
update: function(){
   $.ajax({
     url: '/groups/order_links',
...

...
update: function(){
   $.ajax({
     url: "#{order_links_groups_path}",
     ...

...
     url: "#{order_links_groups_url}",
     ...

...
                   PUT      /groups/:group_id/links/:id(.:format)      links#update
                   DELETE   /groups/:group_id/links/:id(.:format)      links#destroy
order_links_groups POST     /groups/order_links(.:format)              groups#order_links
            groups GET      /groups(.:format)                          groups#index
                   POST     /groups(.:format)                          groups#create
         new_group GET      /groups/new(.:format)                      groups#new
        edit_group GET      /groups/:id/edit(.:format)                 groups#edit

resources :groups do
  resources :links
  collection do
    post 'order_links'
  end
end

class GroupsController < ApplicationController

  ...
  def order_links
    params[:link].each_with_index do |id, index|
      Link.where(id: id).update_all(['position = ?',index+1])
    end
    render :nothing => true
  end
  ...