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My guess is that: 3(a) is true. I'm imagining taking a picture of a knot as a plait, and then constructing a convex tube just for it, where the knot mostly bounces up and down toward you and away from you, but bends and crosses where necesary. If I get a concrete construction, I'll fill in specifics --- currently it's just an idea. I would also guess that no single polyhedron will work for all knots. It's an interesting challenge to try to describe special properties of knots associated with a specific polyhedron.

My guess is that: 3(a) is true. I'm imagining taking a picture of a knot as a plait, and then constructing a convex tube just for it, where the knot mostly bounces up and down toward you and away from you, but bends and crosses where necessary. If I get a concrete construction, I'll fill in specifics --- currently it's just an idea. I would also guess that no single polyhedron will work for all knots. It's an interesting challenge to try to describe special properties of knots associated with a specific polyhedron.

To start, think of a planar polygon that is a slightly modified square. In a square, there are trajectories that bounce back and forth nearly perpendicular to two parallel walls, gradually proceeding down the length. When they reach the end, they glance across a wall and start a journey back.

Update: See Joseph O'Rourke's answer for pictures.

To start, think of a planar polygon that is a slightly modified square. In a square, there are trajectories that bounce back and forth nearly perpendicular to two parallel walls, gradually proceeding down the length. When they reach the end, they glance across a wall and start a journey back.

Added: Plan for realizing all knots with one polyhedron

On further reflection (ha ha), my guess is that we can construct a single polyhedron that will realize all possible knot types. I will sketch the plan --- others are welcome to join in to either fill it in, or refute it.

To start, think of a planar polygon that is a slightly modified square. In a square, there are trajectories that bounce back and forth nearly perpendicular to two parallel walls, gradually proceeding down the length. When they reach the end, they glance across a wall and start a journey back.

Now modify the square by beveling one corner at a 45 degree angle, so that when trajectories reach the cutout, they strike at an angle just less than 45 degrees from perpendicular, so they turn nearly 90 degrees and start in the nearly perpendicular direction, with a slight drift away from the nearby side. Trajectories bounce back and forth for a time between the two major directions, hitting the bevel every other turn, until they reach the end of the bevel, when they start a journey crisscorssing between the second pair of parallel sides.

Put another 45 degree bevel at the opposite corner, so now there are periodic trajectories with this qualitative description with arbitrarily many crossings between pairs of parallel sides.

I have a strong hunch that any knot type can be arranged in a form that has projection to one of these trajectories, just making use of the main weaving part for creating the knot type, but I haven't thought through the details. Let's suppose so, or at least suppose we have a knot that has a weaving style projection.

Now, make a cylinder with the hexagon as above as base, adding two new faces perpendicular to the faces of the cylinder. It is best to visualize it as a thin plate, although logically it is irrelevant how high it is. Trajectories in the cylinder map to trajectories in the hexagon. For each closed trajectory $\gamma$ in the hexagon and for each integer $N$, there is a circle's worth of trajectories in 3 dimensions that project to $\gamma$ and hit the floor and ceiling exactly $N$ times (at regular intervals). The extra circle parametrizes their phase.

Lemma : for any $\gamma$ and any specification of crossing information, there is a billiard trajectory that projects to $\gamma$ and has the given types of crossings.

I suspect this is probably reflexive knowledge to some group of mathematicians. It's saying that if you take a collection $H$ of numbers in the interval (the vertical heights), then look at their orbit under the semigroup $\{ N x + b\}$ generated by translations and multiplication by integers, then the ordering of Sawtooth(Nx + b)(H) can be anything. I think a stronger fact is true the images should be dense in the set of $|H|-tuples$ in the interval.

If this plan holds up, then every knot type can be realized in this 8-faced polyhedron.