When wearing tzitzit with tekhelet, do the same halachot apply to tekhelet as to white regarding if one string breaks off – e.g. my one tekhelet string (tekhelet per Raavad i.e. 2 out of 8) has fallen off completely after the last knot. If all white strings, I would continue to wear this tzitzit lechatchila. As it is a tekhelet string that has broken off, does this make a difference?
Regarding the specific case of the one full blue string being broken on both sides up to the knot. The rule for this would be the same as for white (the details are specified in the Shulchan Aruch O.H. 12). That is: l’chathila one should have all the strings 12 thumb-breadths, 4 thumb-breadths wrapped (gdil) and 8 thumb-breadths hanging (anaf); bedi’avad one must have “k’dei aniva – (KA)” (enough to tie a knot) – less than this and the strings are pasul.
This KA measurement is required for all 4 strings, such that, if you were careful to always keep the four ends on one side of double knots and the opposing ends on the other side, so if even all 4 ends on one side were cut to the knot, but on the other side the 4 strings have KA, so the tzitzit are still kosher; but if 1 string on each side of the knot is cut to the knot, such that one could assume that they are two ends of the same string, then the tzitzit are pasul. So too, if one was not careful to keep the 4 string ends separate each time he made a double knot, or the tying method didn’t include double knots, such that one doesn’t know which string end corresponds to another string end, then the minute two string ends are less than KA, the tzitzit are pasul.
In your case, there is only one blue string and so it is clear that the two ends correspond to one and the same string, and since both sides are lacking KA the tzitzit are pasul. HOWEVER, the Shulchan Aruch (O.H. 12:3) brings two opinions on this issue: according to Rashi (and this is the general halacha) the KA is measured from the end of the wrappings (gdil) – i.e., the hanging strings part (anaf) must be KA; nevertheless, he bring the opinion of the R”Y who holds that we measure the KA from the top knot of gdil next to the begged and so only require KA of wrappings (gdil) – and, concludes the Shulchan Aruch, this opinion can be relied upon in an emergency. The Mishna Berura explains that one should not say a bracha on such a talit and one cannot wear it out in the reshut harabim on Shabbat.
Two more points:
(1) The issue of broken strings is known as “gardumin” and is discussed in Menachot 38b. One opinion is like that poskined in the Shulhan Aruch (O.H. 12:1), as mentioned, that all the strings must be a minimum of KA. The opposing opinion expressed in the Gemara holds that either all the tekhelet must be the full (8 thumb-breadth gdil) length or the white must be the full length, but if both are cut down, then the tzitzit are pasul. This more stringent opinion is held by R. Tam in Tosafot and is brought by the Shulchan Aruch as a stringent opinion that should be followed when possible, however the Rama himself in fact posikins according to R. Tam l’halacha.
Now the Gemara’s discussion distinguished between tekhelet and white, however the Shulhan Aruch and Rama apply it to white only by using the proportions that R. Tam held for tekhelet versus white. That is, he held that there should be two full tekhelet strings and two full white strings; thus his opinion when applied to all white strings is that if more than 2 strings are only KA, then the tzitzit is pasul (as opposed to the Shulhan Aruch who would allow all the strings to be KA).
(2) The issue of how long is KA is also not simple. The Gemara Men. 38b does not resolve the inquiry. Rosh (7) holds the minimum length to be a d’oraita issue and as such we must act stringently – in this case use the longer measure proposed by the Gemara of “tying a slipknot”. Rif and Rambam apparently held the minimum length to be a d’rabanan issue and as such allow one to act leniently – thus in a case of doubt they would permit ANY length to be valid – but this not the halacha (B.Y. O.H. 12:1).
– Mois Navon.