Day length is most important... over 14 hours of light they grow only feeder roots and vegetation, less than 10 they only fatten tubers, gradually moving from one phase to the other in the middle ground.
I read this in a published agricultural college research paper about rooting cuttings and it appears to be the case in my experience, so I try to keep them in the ground for as much photosynthesis as possible during sunny short days. I will protect them if I see one or two cold nights that are predicted to be followed by a week of mild/warm weather, otherwise I'll let them get hit with a frost and dig them up 3-10 days later.
I keep that window because I've heard cold ground helps eyes develop, making dividing before storage easier, but I've been unable to substantiate that piece of wisdom with any published research.