Non-Simple knots in Contact 3-Manifolds

Abstract

We present new families of examples of non-simple prime Legendrian and transversal knots in tight Lens spaces, which demonstrate that the botany of Legendrians in Lens space is rich. In fact, there are more non-isotopic Legendrians that are topologically isotopic to the $n$-twist knot in a Lens space $L(\alpha ,\beta )$ than in $S^3$. We also include connect sum formulas for rational variants of classical invariants, ${\mathrm{tb}}_{\mathbb{Q}}$, ${\mathrm{rot}}_{\mathbb{Q}}$, and ${\mathrm{sl}}_{\mathbb{Q}}$, which indicate that prime knots are the right playground to look for exotic behaviour.

arXiv comment19 pages, 8 figures. Changed to notation to match earlier literature. Comments welcome!