Can the police force you to take a polygraph (lie detector) test?

The police cannot force you to take a polygraph. In fact the police cannot even compel or force you to even talk to them under the Fifth Amendment right against self incrimination.  The best practice is not to talk to them at all.

The Constitution prevents this.  They cannot force you to do anything, unless a judge orders it (like take blood).  You don't even have to talk to them. Remember "you have the right to remain silent."  And that includes lie detector tests and written statements.  You cannot be forced to give evidence against yourself; that's the state's job.  And you should call a lawyer right away.