Shadows, ribbon surfaces, and quantum invariants

TL;DR

This paper extends the understanding of quantum invariants for ribbon links and knotted graphs in complex 3-manifolds, providing new bounds on ribbon genus using Turaev shadows and analyzing the Jones polynomial in these contexts.

Contribution

It generalizes Eisermann's theorem from links in S^3 to colored knotted graphs in connected sums of S^2×S^1, and establishes new bounds on ribbon genus via quantum invariants.

Findings

Jones polynomial divisibility for ribbon links extended to complex 3-manifolds.

Lower bounds on ribbon genus derived from poles of the Kauffman bracket at q=i.

Constructed examples where bounds are sharp and arbitrarily large.

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 $g\geqslant 0$ 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.