In the past I have tried several Joomla! templates. In of these templates the K2 breadcrumbs did not appear. While the breadcrumbs module worked for all other modules the K2 tools modules did not show anything.
I had this problem again while testing a template from Joomlart that is based on the T3 framework. Here it happened again: the K2 tools modules did not display any breadcrumbs. After some research I found the solution in the K2 Forum: the reason for this problem is simply a typo in the definition of the position where the breadcrumbs should appear. Instead of
type="modules" the definition contained
type="module". The problem was “only” caused by the missing “s”.
For the T3 framework version 2 the file
./plugins/system/jat3/base-themes/default/blocks/navhelper.php must be edited and the definition for position “breadcrumbs” must be corrected.