Large Tangle Yields Weakly 4-Connected Minor¶

Claim/Theorem¶

Let \(M\) be a matroid with a tangle \(\mathcal T\) of order \(k\ge 4\) and breadth \(l\). Then \(M\) has a weakly \(4\)-connected minor \(N\) with a tangle \(\mathcal T_N\) of the same order \(k\) and breadth \(l\), such that \(\mathcal T\) is the tangle induced by \(\mathcal T_N\).

Using [[tangle-order-equals-branchwidth.md]], this implies:

if \(M\) has branchwidth at least \(k\ge 4\), then \(M\) has a weakly \(4\)-connected minor \(N\) with branchwidth at least \(k\).

Here weakly \(4\)-connected means:

\(M\) is \(3\)-connected, and
for every partition \((X,Y)\) of the ground set with \(|X|,|Y|>4\),

\[ \lambda_M(X)\ge 3. \]

So high branchwidth can be concentrated inside a nearly indecomposable minor, after peeling away low-order separations.

Dependencies¶

[[tangle-order-equals-branchwidth.md]]

Conflicts/Gaps¶

Weakly \(4\)-connected still gives only a constant lower bound on cut rank across sufficiently nontrivial cuts.
The theorem produces a minor, not necessarily the original code or stabilizer presentation.
For Conjecture 3, this means large intrinsic width cannot be blamed purely on recursive low-order sums, but it still does not approach a linear balanced-cut lower bound.

Sources¶

10.37236/12467
10.1016/j.jctb.2007.10.008