Quantum Topology

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Volume 8, Issue 2, 2017, pp. 249–294
DOI: 10.4171/QT/90

Published online: 2017-05-29

Shadows, ribbon surfaces, and quantum invariants

Alessio Carrega[1] and Bruno Martelli[2]

(1) Università di Pisa, Italy
(2) Università di Pisa, Italy

Eisermann has shown that the Jones polynomial of a $n$-component ribbon link $L\subset S^3$ is divided by the Jones polynomial of the trivial $n$-component link. We improve this theorem by extending its range of application from links in $S^3$ to colored knotted trivalent graphs in $\#_g(S^2\times S^1)$, the connected sum of $g\geqslant 0$ copies of $S^2\times S^1$.

We show in particular that if the Kauffman bracket of a knot in $\#_g(S^2\times S^1)$ has a pole in $q=i$ of order $n$, the ribbon genus of the knot is at least $\frac {n+1}2$. We construct some families of knots in $\#_g(S^2\times S^1)$ for which this lower bound is sharp and arbitrarily big. We prove these estimates using Turaev shadows.

Keywords: Jones polynomial, shadow, ribbon surfaces

Carrega Alessio, Martelli Bruno: Shadows, ribbon surfaces, and quantum invariants. Quantum Topol. 8 (2017), 249-294. doi: 10.4171/QT/90