Lattice Tadpoles

We prove several rigorous results about the asymptotic behaviour of the numbers of tadpoles (or lassos) embedded in a lattice, including cases where the head is subject to a constraint like being unknotted, or where the tail pierces the surface spanned by the head. Similar results can be proved for other homeomorphism types such as dumbbells, twin tailed tadpoles and two tailed tadpoles.