Is it possible to calculate the position of touch on a conductive thread?
If you put the thread across the terminals of a meter measuring resistance, no matter where you touch the thread there should be a small change in resistance but, unfortunately, that resistance change would be the same throughout the length of the thread - this on its own prevents you from achieving what I believe you want.

However, if you have two threads very close to each other but not touching you should be able to roughly transfer the voltage at the contact point to the 2nd thread. In effect, the 2nd thread becomes the wiper of the potentiometer but feed that 2nd thread into a high impedance amplifier or the DC voltage that your skin transfers will be severely diminished.

Alternatively use an AC signal across the 1st thread and therefore transfer the ac signal to the "wiper" thread - this will be more reliable I suspect.