Tony in the US is often short for Anthony.
Chris, Kris, or Cris can be short for Christina, Kristina, Kristofer, Christopher, Kristian, Kristian, Christy, Krista, or Kristi. Sometimes you'll see Crys, Chris, Krys, Kris, or Cris for Crystal/Cristal/Chrystal/Krystal.
You'll also sometimes see Christopher shortened to Topher or to Kit.
These two are incredibly common in the UK. You will occasionally see Ant (as in And & Dec TV presenters) but almost no one uses Anthony, they are usually all Tonys.
The female short names is usually spelt Toni, but pronounced the same.
'Antony' is also related to 'Antonius' (as is 'Italian Antonio').
One weird one (for me) from English is 'William' -> 'Bill' since there's no 'B' in 'William', and no 'W' in 'Bill'. I wonder what's happening there.
Chris can be short for Christian.