Quantum deformations of fundamental groups of oriented 3-manifolds

Abstract

Abstract. We compute two-term skein modules of framed oriented links in oriented 3-manifolds. They contain the self-writhe and total linking number invariants of framed oriented links in a universal way. The relations in a natural presentation of the skein module are interpreted as monodromies in the space of immersions of circles into the 3-manifold.

An intrinsic approach to invariants of framed links in 3-manifolds

“…Also the presence of framing requires an adaptation of the arguments: to formulate the correct "global integrability condition" (equation ( 6) below) we need a notion of global framing around homotopies of links. The definition of such a notion is facilitated by the works of Chernov and Kaiser [3], [13] (Definition 2.7). For arguments that are very similar to these in [14], [16] we will refer the reader to these articles for details.…”

“…To prove (6) we will turn our attention to ordered links: First we note that the invariant f pulls back to an invariant on z L .1/ via the forgetful map r. After iteratingˆseveral times if necessary we can assume that it lifts to a loop in M L .P; M / based at L (compare p. 3874 of [13]). Given a self-homotopyˆof CL and the associated quantity Xˆ, liftˆto a closed homotopy ‰ in M L .P; M / and let X ‰ denote the lift of Xˆ.…”

“…The two framings might differ by twists on the components of CL but the total singed number of the twists must be zero. The total sum of such twists is captured exactly by the quantity f( compare, Theorem 6 of [13]). The framing of CL lifts to one on L and going around the self-homotopy of L that liftsˆalso preserves the homotopy class of the framing vector field.…”

“…In particular, Przytycki [22] introduced a two term homotopy skein module of framed links in oriented 3-manifolds as quantum deformation of the fundamental group. In [13] Kaiser calculated this module over the ring of Laurent polynomials with Z-coefficients. He showed that if a 3-manifold contains no non-separating 2-spheres or tori then Przytycki's module is a symmetric algebra of the free module with basis the set of non-trivial conjugacy classes of 1 .M /.…”

“…Kaiser also studied several variations of two term skein modules and put the classical self-linking number for null homologous knots as well as Chernov's generalization of it in the skein module theory framework. For details the reader is referred to [13].…”