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 3, but fails for k 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 τ has an internally 4-connected minor whose unique 4-tangle lifts to τ.