Twist spinning knotted trivalent graphs (Q2790213)

A knotted trivalent graph (KTG) is an embedding into 3-space \(\mathbb{R}^3\) of a graph satisfying the condition that every vertex has degree 3. A knotted sphere is a smooth embedding into \(\mathbb{R}^4\) of a 2-sphere. A knotted 2-dimensional foam is analogous to a knotted sphere in the same way that a knotted trivalent graph is analogous to a knot. A 2-dimensional foam in \(\mathbb{R}^4\) is unknotted if it is ambient isotopic to an embedding in \(\mathbb{R}^3\), while a knotted graph in \(\mathbb{R}^3\) is unknotted if it is ambient isotopic to an embedding in \(\mathbb{R}^2\).NEWLINENEWLINEIn this paper, the authors define a knotted 2-dimensional foam and show that there is a non-trivial foam obtained by a \((\pm 1)\)-twist spinning of a knotted \(\theta\)-curve: this result implies that a \((\pm 1)\)-twist spun KTG is not always unknotted. The non-triviality is shown by giving the foam a non-trivial coloring by the associated quandle of a \(G\)-family of quandles. Further, the authors show that for an unknotted graph or an almost trivial graph, its \((\pm 1)\)-twist-spin is unknotted. Here, a graph in \(\mathbb{R}^3\) is almost trivial if it is a non-trivial graph such that every proper subgraph is unknotted.