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Lift in the 3-sphere of knots and links in lens spaces
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abstract
An important geometric invariant of links in lens spaces is the lift in the 3-sphere of a link $L$ in $L(p,q)$, that is the counterimage $\widetilde L$ of $L$ under the universal covering of $L(p,q)$. If lens spaces are defined as a lens with suitable boundary identifications, then a link in $L(p,q)$ can be represented by a disk diagram, that is to say, a regular projection of the link on a disk. Starting from a disk diagram of $L$, we obtain a diagram of the lift $\widetilde L$ in the 3-sphere. With this construction we are able to find different knots and links in $L(p,q)$ having equivalent lifts, that is to say, we cannot distinguish different links in lens spaces only from their lift.
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Link in $\mathbb{R}\mathbb{P}^3$ and the Topological Vertex
First topological-vertex computations of colored unknot and Hopf link invariants in RP^3 yield power series with positive integer q-expansions, conjecturally the Poincare series of a new link homology.
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