One of the strings on my tzitzit tore and It was the white end of my techelet string (I wear Rambam techelet). I was wondering if I could tie that string together with another string and apply glue to the knot to make a longer sting?

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.

By the way, if it were a Raavad or Tosfot type string where there was blue on both sides of the hole, then one could apply the ruling of the Mishna Berura that only one side of the strand needs to have the “kdei aniva” length. That is, if only one SIDE of one strand was torn (even less than kdei aniva) then the strand (as a whole) is still considered kosher and so one could adjoin an extra bit of string to bring it to the l’chathila length (i.e., twice the length of the gdil). The remedy the Mishna Berura mentions is specifically a “permanent knot” and so I would advise using a knot and then, if so desired, applying glue as an extra precaution (but not to depend on the glue).

– Mois Navon.
P.S. a thumbreadth is 2 cm (acc. R. Na’eh) or 2.4 cm (acc. to the Hazon Ish). “kdei aniva” is two thumbreadths (agudalim) – either 4 cm or 4.8 cm.