TL;DR
This paper rigorously analyzes the asymptotic behavior of lattice tadpoles and related structures, considering various constraints and topological types, advancing understanding of their combinatorial properties.
Contribution
It provides new rigorous results on the asymptotic counts of lattice tadpoles and similar structures with different topological constraints.
Findings
Asymptotic formulas for tadpole counts under various constraints
Extension of results to other homeomorphism types like dumbbells and twin tailed tadpoles
Rigorous mathematical proofs of these asymptotic behaviors
Abstract
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.
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