Agreed with contractor's agent for work specified in a crude drawing for $19,000. $14,500 was paid but balance of work is disputed between parties. Do they have a legal right to file a mechanics lien?
That depends on additional factors. Yes a contactor can record a mechanic's lien without a written contract, UNLESS the work was done on an owner-occupied residence. In that case, a written contract is required between the owner and the contractor who is recording the lien. The contractor must have also mailed a valid and timely preliminary 20-day notice. No preliminary 20-day notice, no lien rights for the contractor.
There are other requirements at play, but I would need more details and the two I have described above are the big ones.