I had cross-posted this on GIS.Stackexchange, and I got an answer there.
Basically, I was going about it the wrong way. Instead of setting the timeExtent on the map, I needed to set the time slider to a specific position, and the map will be automatically updated to that time extent.
To do this, I just needed to set the ThumbIndex on the slider, using the following code:
// Assuming time slider has a single time thumb timeSlider.setThumbIndexes([4]);
I had cross-posted this on GIS.Stackexchange, and I got an answer there.
Basically, I was going about it the wrong way. Instead of setting the timeExtent on the map, I needed to set the time slider to a specific position, and the map will be automatically updated to that time extent.
To do this, I just needed to set the ThumbIndex on the slider, using the following code:
// Assuming time slider has a single time thumb timeSlider.setThumbIndexes([4]);