While tying my tzitzit according to the Rambam Teimani method, the techelet string broke (high up on the blue though there is still enough string to tie several chulyot.
The answer to this question is just about the same as to the question if the string broke after the whole tzitzit was tied. I will reiterate that answer and relate to the specific issue of having tied even only one hulya.
The Mishna Berura (12:1:7) discusses a similar case and says that if one has a kosher string (i.e., only one side being too short), you can tie on the extra strand to make the full length.
First of all, lets be clear on terminology: the tzitzit are made of 4 full strands that are folded over through a hole in the corner of the Talit, making two sides to each full strand. The tzitzit are then knotted, making a section known as the gdil, and leaving a hanging free section known as the anaf.
The Shulhan Aruch explains that for tzitzit to be kosher (bidieved) all four strands must have at their anaf hanging at least “kdei aniva” (the amount needed to tie a knot) on one side (i.e., one side of strand could be cut to the edge of the bottom knot of the gdil as long as the other side of the strand was kdei aniva). However, if one strand’s anaf was cut to the point that it did not have a length of kdei aniva on either side of its hanging anaf, then the strand is pasul and so is the whole corner of tzitzit. Thus, for a strand to be kosher at least one of the sides of the strand that hangs below the last knot of the gdil must be the length of kdei aniva.
However, I hesitate to apply this ruling here because the Mishna Berura is talking about an all white string. In the case of a Rambam tekhelet string, one “half” is blue and the other “half” is white, so it seems that the white side of the string is not considered the completion of the tekhelet side. If this is true, then you have invalid tekhelet and fixing it while still tied to the talit would be considered doing it “lo min ha-asui” (not in it usual way) and thus forbidden. So really you need to untie the corner, take the strings out of the talit, tie the blue pieces of string together and then restring them
in to the talit and tie them.
Now in your case, you have one kosher hulya and quite a bit more than “kdei aniva” blue string hanging. However this hanging string is BEFORE a full gdil is tied. The Biur Halacha (12:1, s.v. “shema nifsak”) says that it would seem tying one hulya and knot, which makes the tzitzit minimally kosher d’oraita, would suffice to allow one to tie on to the broken string – however he rejects the idea because one hulya was not the intent of tying the tzitzit. So really,after a tear, you would need to have enough string to tie minimum kosher d’rabanan tzitzit which is 4 thumbreadths of gdil and 8 thumbreadths of anaf (in total quite a bit longer than 12 thumbreadths because the gdil consists of wraps and knots).
As such, if you don’t have over 12 thumbreadths hanging, I would advise untying the corner, attaching the two torn strands together by knotting and then maybe glue, and then retying the corner.
– Mois Navon.