Is knuckle-walking efficent on snow / ice / thick snow ?
Can a predator large as a polar bear evolve this? Is protecting the claw in such way, to far fetch considering the fact that the beast need to travel long distance in the snow covered tundra and ice sheeet in order to track its food?
It doesn't seem a good idea.
Knuckle-walking concentrates the weight of the animal on the rather limited areas of the knuckles, meaning that both on snow or ice this will give problems: to avoid sinking in snow you want to spread your weight on a large area so that the snow doesn't collapse too much (principle on which snowshoes are based), while on ice the smaller the area the less grip you have, and knuckles do not even allow using claws/nails for additional grip.
The only animals that do knuckle walk have relatively delicate, and specialized hands that interfere with normal walking. You have Apes with dexterous fingers that can't easily stand upright for long distances and anteaters with specialized digging claws that are too long an cumbersome to be moved out of the way to put weight on the sole of the foot.
Claws are pretty durable. They're designed to tear apart trees, dig through soil, and rip through flesh. Bear claws don't need protection, but lots of things need protection from bear claws.
Keep in mind that polar bears have evolved to travel hundreds of miles in search of food, across snow covered tundra and ice sheets. Knuckle walking seems like an added complication for no benefit.
