THE (2, ∞)-SKEIN MODULE OF WHITEHEAD MANIFOLDS

Abstract

To any orientable 3-manifold one can associate a module, called the (2, ∞)-skein module, which is essentially a generalization of the Jones polynomial of links in S3. For an uncountable collection of open contractible 3-manifolds, each constructed in a fashion similar to the classic Whitehead manifold, we prove that their (2, ∞)-skein modules are infinitely generated, torsion free, but not free. These examples stand in stark contrast to [Formula: see text], whose (2, ∞)-skein module is free on one generator. To each of these manifolds we associate a subgroup G of the rationals which may be interpreted via wrapping numbers. We show that the skein module of M has a natural filtration by modules indexed by G. For the specific case of the Whitehead manifold, we describe its (2, ∞)-skein module and associated filtration in greater detail.