Shadows, ribbon surfaces, and quantum invariants

Abstract

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 $g⩾0$ 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.