Surface irregularities (bumpiness) are the reason why lower pressure tyres are now preferred on bikes. The idea is that a very rigid tyre will deflect the bike and rider up and down which wastes some energy/momentum as well as fatiguing the rider, whereas a lower pressure tyre can absorb those irregularities and "roll-over" the bumps. This is part of the reason that recent thinking has moved from skinny high pressure tyres, to wider medium pressure tyres. (Wider tyres will tend to roll quicker than a thin tyre at the same pressure - something to do with how the contact patch deforms the rubber).

However, cycling tracks are designed to be very smooth which is why high pressure tyres are still used there.

Any bump results in some energy transfer. In the case of small enough bumps and tires at ideal pressures, most energy is returned, but not all. These losses accumulate. The question is "how much does it add up to?" This is why I recommend using the phrase "negligible effect" instead of "no effect".

You're not going up and down the track during an hour record. Just doing laps at the bottom (zero elevation change). Track surfaces aim to be very smooth in general.

> You're not going up and down the track during an hour record.

Here the English language obscures the physics. Sure, the black line on the track is at a constant elevation. But the tire's point of contact is different from the system's center of mass (CoM). CoM is key here. When a rider tilts in the turns, the CoM lowers. In the straights, it raises. So, you _are_ going up and down during the hour record.

The question now becomes: how much effect does this elevation change have?

It is one thing to be aware of the effect, run the calculations, and find the result is negligible. Has anyone done this? That would be an interesting analysis, and I'd like to see it.

With this in mind, I will make another claim: for a particular rider, there is an ideal line around a velodrome that would minimize center-of-mass elevation change. This line would be faster than the current black line. How much faster? This would be a fun simulation problem.

Another interesting connection: center of mass and bicycling explains why pumping works on a BMX track, a pump track, a trail, and so on. (There are other mainstream explanations, but I think the CoM explanation is the most elegant.)

It’s an interesting question & thought experiment! To the degree that it even matters compared to all the other bigger forces, I would put money on riders naturally adjusting for CoM changes by riding slightly higher during the turns; I’d bet they already take approximately the ideal racing line you suggest. I just watched a couple of velodrome rides on YouTube, and it does seem like riders are often closer to the red line during the turn and the black line during the straightaway, statistically, but it’s noisy and would need to be measured.

The CoM’s elevation change on a velodrome track is due to roll rotation around the direction of travel, not to climb & descent. You can’t pedal harder to recover from a lean, so this is a different kind of up and down than straight line elevation changes. It makes sense that work is being done somehow if the CoM moves up and down, but the turns come with necessary changes to the higher moments of inertia anyway that flattening the CoM elevation doesn’t change. I’d speculate that the ideal CoM line might not be flat, in the presence of mandatory high speed banked turns; the fastest line and the line minimizing CoM elevation change might be two different lines. Do also keep in mind that on a velodrome track, a higher elevation line is a slightly larger radius turn & longer travel path. It’s also possible that trying to compensate for CoM elevation change adds as much time as it saves.

Wouldn't you mainly get the energy back? Like, you use some going up the wall, and then gain it all back going lower.