Every k-tangle in a graph is the lift of a k-tangle in a topological minor of size bounded by a function of k, reducing the vertex-set induction problem to bounded-size graphs.
Characterising 4-tangles through a connectivity property
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abstract
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 $\tau$.
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On vertex sets inducing tangles
Every k-tangle in a graph is the lift of a k-tangle in a topological minor of size bounded by a function of k, reducing the vertex-set induction problem to bounded-size graphs.