Tait′s Flyping Conjecture for Well-Connected Links

We call a diagram of a link well-connected if it is connected, has no 2-edge cuts, and the only 4-edge cuts are those made by a crossing. We prove Tait′s flyping conjecture for well-connected diagrams, i.e., any two well-connected alternating diagrams represent equivalent (= ambient itotopic) links, if and only if these diagrams are the same up to trivial operations.