Every large $k$-connected graph-minor induces a $k$-tangle in its ambient graph. The converse holds for $k\le 3$, but fails for $k\ge 4$. This raises the question whether `$k$-connected' can be relaxed to obtain a characterisation of $k$-tangles through highly cohesive graph-minors. We show that this can be achieved for $k=4$ by proving that internally 4-connected graphs have unique 4-tangles, and that every graph with a 4-tangle $\tau$ has an internally 4-connected minor whose unique 4-tangle lifts to~$\tau$.

翻译：每个大型$k$-连通图子式都会在其环境图中诱导出一个$k$-纠缠。当$k\le 3$时，逆命题成立，但对于$k\ge 4$则不成立。这引发了一个问题：能否放宽"$k$-连通"条件，从而通过高度凝聚的图子式来刻画$k$-纠缠？我们证明对于$k=4$，这一目标可以通过证明内部4连通图具有唯一的4-纠缠，且每个包含4-纠缠$\tau$的图都存在一个内部4连通子式，其唯一的4-纠缠可提升至$\tau$来实现。