It's further worth noting that I do not claim by any stretch that European axes are inherently superior to American axes, and I own and use polled American and American-styled axes all the time. I simply prefer to use my Italian axes for most of my axe tasks these days because I personally find the particular blend of performance features and functional prioritizations to be pleasant to use for my particular applications. The primary motivation I have for defending them stems from a personal belief that they are somewhat unfairly snubbed when their oft-touted shortcomings can be accounted for in application, technique, and/or handle tuning, and that they do offer a range of styles that may be beneficial to some users for certain contexts of use vs. more common market offerings. American axes make some tradeoffs of their own, and such is the case with any tool.
Polls give the advantage of making unified-axis handles easier to make and (if the poll is of sufficient proportion) to allow the unified region to run right up to the head. They also lend the tool better pounding ability (on non-ferrous targets only, unless the poll is hardened) because you have a thicker and more robust "hammer face" there. The disadvantage is that if bit shape is held fixed, the head is now heavier for a given bit dimension. If weight is held fixed, the bit geometry and/or bit-to-eye transition has to be altered to permit the eye to be closer to the edge.
A lack of a poll allows for a larger bit (either wider, deeper, thicker, or a mix of the three) for a given head weight, as well as allowing for a deep bit without the edge sitting so far forward of the tool's axle in use. That is to say that because the axle runs through the center of gravity, a distance of 5" from the edge to the axle on a polled axe with the center of gravity in the middle of the eye might be only around something like 3" if the poll were eliminated, which will make for less magnification of wobble for the particular bit depth. The disadvantage is that to have a handle laying along a unified axle, you'd need to use a larger starting piece of wood and have better grain alignment because of the offset you'll be putting in the neck, and there'll be a few inches of handle even with that offset that won't be along that unified axle, because the more severe the radius of the offset the weaker that bend will be. However, setting the eye that far back from the edge allows for a larger eye without interfering with the action of the bit in use. This is of advantage with many European axes because they do not natively have access to hickory, and the norm is to use ash or beech. Some regions don't even have those in abundance and have to use woods like birch. The further back from the edge you go, the wider the umbra of the bit geometry is, so there's more lateral room for a wider eye without it getting in the way of things, allowing weaker woods to be used satisfactorily for handles. This is of even greater advantage when using slip-fit handles because they are limited in their handle shape by what can pass through the eye of the axe. More on that in the next paragraph.
Now, wedged handles have the advantage of being able to have pretty much whatever shape the user desires so long as it can be made sturdy enough for the application. This allows for handles to have a large knob on them, and the eye may be made smaller so it can be placed closer to the bit without bulging the geometry out. Doing so requires stronger wood (enter hickory) to hold up, however. And as we all know, wedged handles have provided so much of an issue in satisfactorily keeping them affixed that all manner of inventions have been made for the purpose of doing so, including the dreaded PermaBond, let alone the use of screws, nails, and gawd knows what else that many have (in a breach of good practice) resorted to driving into them to tighten them up when loosened. People sometimes split handles in the process of wedging them, and so on and so forth. While there are some methods that can be taken, a broken handle in the field without access to an assortment of hand tools presents serious challenges, as well; a drill, vise, hammer, and saw being just shy of necessities for making and fitting a new handle. And if you hang a head crooked (as factory hangs can easily be) you're kind of stuck with it unless you're able to withdraw the wedge without damaging the surrounding wood in the process (there are techniques for this that can work quite well, but they aren't always successful.) However, the freedom of form they offer is of great appeal, and particular care and diligence in the hanging and wedging process is usually able to give solid and lasting results.
I often remark that a slip fit eye offers the advantage of allowing one head to wear many handles (really quite a handy feature) but perhaps the largest benefit is the ease of making and replacing handles for them with limited tools, which is a major reason why they are popular in Latin America, especially amongst indigenous groups that use very few manufactured items. The head alone can be used in the entire process of making its own handle, and at some point (when I actually have time and there's not two feet of snow on the ground) I plan on doing just that by means of demonstration. Broken handles are easily extracted from the eye by simply driving them out. It is also impossible for the head to go flying off the end of the handle. However, they do impose limits on the dimensions of the handle, making the size of any knob restricted to a size and shape that can pass through the eye. It's possible to make a second piece of wood that slides onto the end of the handle and is affixed by a tapered pin, thus allowing a knob to be present and still rapidly disassembled, but that gets you back into the territory of needing more complicated tools. Also, because of the means by which they are affixed, slip-fit style eyes can usually be made shallower than wedged ones without causing problems in use, though of course a deeper eye will always make the head/handle bond better. If the wood ever shrinks as a result of arid weather, the handle can simply be tapped down a little more into the head and it's nice and tight again.
Lastly, pertaining to bits, the bit can be generally broken down into the categories of thickness, width, depth, and the bit-to-eye transition. The force of the blow is divided amongst penetration, wedging, and length of cut, and it is considered desirable for the blow to go as deep and wide as possible with each blow without becoming stuck or difficult to withdraw. Penetration can be increased by thinning and/or narrowing the bit. A thinner bit will bite deeper, but the width of the chip it can effectively pop will be reduced (as well as reducing splitting ability) while a narrower bit can retain chip-popping ability but will then require more blows to connect across the face of a log. A wider bit will reduce penetration but also reduce the number of blows required to connect across the log face. A thicker bit will reduce penetration, but increase the maximum width of the chip that it will throw and increases general splitting ability. The bit-to-eye transition is important in deep cuts, where the eye can glance off the mouth of the notch, or in splitting where too sharp of a transition can slam the brakes on the cut like hitting a wall. A balance of these features impacts the potential best applications of use for the particular head, and getting the right combination of them can make the difference between a pleasant or hellish time. One of the criticisms I often lay on the popular Cold Steel Trail Hawk, for instance, is that the geometry is both very thin and extremely narrow, with a sharp bit-to-eye transition, and that makes chopping with it like pecking away with a woodpecker's bill. It's fine for a weapon, but not the best for a field tool.
Additional factors also come into play with bits, such as curved vs. straight edges, where more curved bits are more penetrating but make connecting blows more difficult due to the scalloped bottom they're predisposed to leaving, while flat edges bring more shock in use or are otherwise prone to land on their heel or toe before the rest of the bit, but are easier to connect blows with and make clean cuts, which is why they're favored on carpenter's axes. I personally tend to prefer just very slightly curved bits, which are just curved enough to give the "guillotine" assist in the chop but are otherwise pretty flat.
In thicker geometries with more wedging force resisting the cut, frictional forces increase and this gives rise to the convex cheeks matter. In such circumstances the pinch of the wood is reduced by minimizing surface contact via the convex cheeks. However, the thinner the bit, the less feasible such a feature becomes, and the less influential, because the wood isn't being forced so greatly out of the way in the first place and so is not pushing back against the bit as much.
All of these features can be mixed and matched in varying degrees, and you can find examples of pretty much every permutation along the spectrum. Even if limiting it to American styles, you can see how these features were altered in varying degrees to optimize their forms to regional contexts of use (as well as some certain amount of aesthetic taste, as well) and functionally it's often possible to arrive at many permutations of equally effective solutions to a problem (or as the saying goes, "more than one way to skin a cat") and matters of preference or slightly varying ranges of intended function can change the optimization. For instance, even if two fellows both need axes to handle tasks A, B, and C, fellow no.1 may be mostly doing task A, with a moderate amount of task B, and just occasionally task C. Fellow #2, however, may be mostly doing task B, a moderate amount of task C, and just occasionally task A. Two different axes optimized for those different prioritized mixes of tasks will likely be pretty similar in most respects, but if both fellows trialed both axes for their contexts of use they'd find the one preferable to the other even if both got the work done effectively. That doesn't make one of those axes inherently superior to the other in a broad sense, but rather for that particular fellow's context of use, one was superior for their particular purposes. Now, it is certainly possible for an axe to simply be poorly designed or executed, but in the case of traditional patterns there's usually sound reasoning behind why they evolved into the forms that they did, and it's usually decoded by sizing up what aspects are emphasized over others, and what techniques would be best applied to the tool, given those particular features.
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