asked a great question about rear suspension linkages that may be of interest to others. To paraphrase, he wants to clarify whether a design with the wheel attached to the chainstay / swingarm (often called a "faux-bar") is four-bar linkage. He further asks how flex-pivot and Horst designs fit in. Below is my response.
It's important to consider the purpose of the design. A four-bar system can decouple the movement of the wheel and brake from the element attached to the frame, which we can refer to as the chainstay or swingarm. Mounting the wheel and brake to this element removes valuable design freedoms offered by a four-bar system. A system with the wheel and brake attached to the swingarm is a single-pivot with linkage-actuated shock.
It's not incorrect to call it a four-bar, but it's misleading. If we allow such a system to be called a four-bar, we could add any number of elements to the shock linkage and call it an n-bar linkage, but it still behaves as a single-pivot with linkage-actuated shock. It would be like saying a line is a one-dimensional object … and also a two-dimensional object that doesn’t use the second dimension, and a third-dimensional object that doesn’t use the third dimension, and so on. While true, these are “trivial cases,” and trivial cases are typically ignored. Therefore, the best description is a single-pivot with linkage-actuated shock.
The Horst design is not the only true four-bar. A dw*link, for example, is the same as a Horst, but with the chainstay shrunken to become the lower link, the “seatstay” expanded to become the entire rear triangle, and the rocker often left unchanged. Same with VPP. Same with KS-Link. And Maestro. And MLink. And Lawwill. And ETS. And … take your pick; they’re all four-bar systems. Even Split Pivot and ABP are four-bars, since the brake is not attached to the swingarm. The ability of Split Pivot and ABP to decouple the braking instant centre from the swingarm pivot makes them more more closely related to a "true" four-bar design than a single-pivot with linkage-actuated shock.
A flex-pivot design with the flex mostly confined to the seatstay is a single-pivot with linkage-actuated shock. As before, the braking and pedaling instant centres are concentric with the swingarm pivot, so those properties behave the same as a single-pivot. Obviously, the linkage modifies the shock actuation, much as it would if there was a rotational element pivot.
A flex-pivot design with the flex mostly confined to the chainstay (ex. Cannondale, Spot), however, is equivalent to a four-bar.