Framed knots in 3-manifolds and affine self-linking numbers

Details Framed knots in 3-manifolds and affine self-linking numbers Framed knots in 3-manifolds and affine self-linking numbers

2001-05-16

arxiv.org/abs/math/0105139v3

J. Knot Theory Ramifications 14 (2005), no. 6, 791--818.

Mathematics

Geometric Topology

This is a major revision in which we make explicit the construction of affine self-linking numbers, that are generalizations o

Scientific paper

The number $|K|$ of non-isotopic framed knots that correspond to a given unframed knot $K\subset S^3$ is infinite. This follows from the existence of the self-linking number $\slk$ of a zerohomologous framed knot. We use the approach of Vassiliev-Goussarov invariants to construct ``affine self-linking numbers'' that are extensions of $\slk$ to the case of nonzerohomologous framed knots. As a corollary we get that $|K|=\infty$ for all knots in an oriented (not necessarily compact) 3-manifold $M$ that is not realizable as a connected sum $(S^1\times S^2) # M'$. This result for compact manifolds was first stated by Hoste and Przytycki. They referred to the works of McCullough for the idea of the proof, however to the best of our knowledge the proof of this fundamental fact was not given in literature. Our proof is based on different ideas. For $M=(S^1\times S^2) # M'$ we construct $K$ in $M$ such that $|K|=2\neq \infty$.

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