Minimally knotted graphs in \(S^ 3\). (English) Zbl 0723.57008
Let G be a planar graph embedded in \(S^ 3\). If the embedding is planar, then G is said to be unknotted. If each proper subgraph of G is unknotted, then G is said to be locally unknotted. It is natural to conjecture that if \(\pi_ 1(S^ 3-G)\) is free and G is locally unknotted, then G is unknotted. In the first half of this paper, the authors prove the above conjecture for various kind of graphs including handcuff graphs, \(\theta\)-curves, and double \(\theta\)-curves. The proof uses some results of combinatorial group theory. Since then, Scharlemann and Thompson proved the above conjecture in full generality.
In the latter half of this paper, the authors give a new method for proving nontriviality of a trivalent graph, by developing an algorithm that starts with a trivalent graph G and produces a trivalent graph \(G_ I\) with fewer vertices than G such that \(G_ I\) nonplanar embedded implies G nonplanar embedded. By this method, they reprove the nontriviality of several minimally knotted graphs (that is, locally unknotted graphs which are nontrivial). It is noted that A. Kawauchi [Osaka J. Math. 26, 743-758 (1989; Zbl 0701.57015)] proved their conjecture that every planar graph has a minimally knotted embedding.
In the latter half of this paper, the authors give a new method for proving nontriviality of a trivalent graph, by developing an algorithm that starts with a trivalent graph G and produces a trivalent graph \(G_ I\) with fewer vertices than G such that \(G_ I\) nonplanar embedded implies G nonplanar embedded. By this method, they reprove the nontriviality of several minimally knotted graphs (that is, locally unknotted graphs which are nontrivial). It is noted that A. Kawauchi [Osaka J. Math. 26, 743-758 (1989; Zbl 0701.57015)] proved their conjecture that every planar graph has a minimally knotted embedding.
Reviewer: M.Sakuma (Osaka)
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57M15 | Relations of low-dimensional topology with graph theory |
Keywords:
spatial graphs; planar graph; handcuff graphs; \(\theta \) -curves; minimally knotted graphsCitations:
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