Which of the following noxious weeds has deep roots and cannot be controlled by digging or tilling up the plants?

The correct answer is Russian knapweed, which is known for its deep and extensive root system that makes it particularly difficult to control using traditional methods like digging or tilling. This perennial plant can regenerate from even small root fragments left in the soil, allowing it to persist and spread despite surface removal efforts.

In contrast, the other options, while they can also be challenging, do not have the same level of resilience. Curly dock, for example, has a taproot that can also regenerate, but it is often more susceptible to pulling or tilling than Russian knapweed. Poison hemlock is primarily biennial and can be managed through various means if detected early. Puncture vine, known for its spiny seed pods, can be controlled effectively with digging or tilling before it sets seed. Thus, Russian knapweed stands out for its deep rooting characteristics and resistance to common control methods, validating its selection as the correct answer.