Characterising 4-Tangles Through a Connectivity Property

Every large $k$-connected graph-minor induces a $k$-tangle in its ambient graph. The converse holds for $k\leqslant 3$, but fails for $k\geqslant 4$. 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 $\tau$.