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A single string tore from the Tzitzis:^[1]

Tore within the Gdil/braid, or beneath the Gdil by the hole?[2]

If even one string tore from the root of the fringes, the Tzitzis is invalid.[3] This means that if even one string tore near the corner hole, within the area that precedes the braid[4], or within any area of the braid[5], the Tzitzis is invalid. [Thus, any tear that occurs beneath the loose area called Anaf, invalidates the Tzitzis.]

One string tore above the Gdil/braid?[8]

If a single string [of the 8 strings] tore in the area of loose strings that extends past the braid, called the Anaf, it is valid. This applies even if it completely tore until the actual start of the braid.[9] [Nevertheless, it is an act of piety to always have 8 full strings of 24 cm, as explained above.]

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[1] Admur12:2-3; Siddur Admur

[2] Siddur Admur; Shlah Chulin 112a in interpretation of Rav in Menachos 39a; Elya Raba 8:10 in name of Taz and Shlah; Emek Haberacha 2:13; Soles Belula 12:2

Background:

The Gemara states that if a string tore in the “Ikkur” or root of the Tzitzis, the Tzitzis is invalid. There are various interpretations of this word “Ikkur”. Does it mean the “Anaf”, the loose strings; does it mean the Gdil, the braided area; does it mean the area below the braid, which is the strings that pass through the hole? Admur in the Siddur, as well as the other Poskim listed above, take the approach that it refers to the braid and below the braid.  In the Shulchan Aruch however Admur takes the last approach, as will be explained. 

[3] Admur ibid; Rav in Menachos ibid

[4] Admur 8:13; Siddur Admur; Taz 8:8; Poskim ibid and in addition so also rule: Kitzur SHU”A 9:13; M”B 8:21; Biur Halacha 12:1 “Yeish Lismoch”; Tehila Ledavid 12:2; Chazon Ish 3:6; Kaf Hachaim 8:37; See Piskeiy Teshuvos 12:3

The reason: As when the tear occurs beneath the Gdil, by the holes, it is considered as if both sides have been cut off, and hence an entire string is missing. [Tehila Ledavid ibid]

[5] Siddur Admur; Shlah Chulin 112a in interpretation of Rav in Menachos 39a; Elya Raba 8:10 in name of Taz and Shlah; Emek Haberacha 2:13; Soles Belula 12:2

The reason: As whenever it tears beneath the Anaf, it is viewed that the entire string has torn and one hence only has three whole strings remaining. [Shalah ibid]

Ruling of Admur in Shulchan Aruch and other Poskim: In the Shulchan Aruch 12:6 Admur rules that only when a string requires a minimum length of 4 cm, is there a difference of opinion regarding if the string tore in the braided area. This implies that if the string does not require 4 cm, such as if the other side of the string still maintains a 4 cm length, then according to all [even Rashi] it does not invalidate the Tzitzis even if it tore within the braid. So rules also Biur Halacha 12:1 “Yeish Lismoch”; Kitzur SHU”A 9:13; Tehila Ledavid 12:2. This is all regarding if the tear occurred within the Gedil. If however the tear occurred below the Gedil, by the hole, then it is invalid according to all. [Admur 8:13; Poskim ibid]

Ruling of Mishneh Berurah: In Biur Halacha ibid, the M”B rules that if the tear occurred within the first set of loops, it is invalid. If it occurred after the first two knots made on top of the first set of loops, and there is 4 cm of string that protrudes, it is valid. [See Piskeiy Teshuvos ibid]

[6] As the last two knots are still considered part of the Gdil from which the Anaf extends, and hence when a tear occurs in that area, the Anaf is considered to have lost one whole string.

[7] As the knots separate between the Anaf and the Gedil, and hence if the string is now extending from beneath the knots, within the Gedil, it is now invalid as it is considered as if its root is torn, and hence both sides are now considered torn. However, according to this understanding, even if a string did not tear but simply unraveled from the knots, and now extends below the knots, it would be invalid, and this is preposterous to even ponder.

[8] Admur 12:2; Siddur Admur; Michaber 12:1 [Up to four strings from one side may tear completely]

[9] The reason: As Tzitzis can only become invalid due to a tear, if an entire one of the four long strings is torn. Hence, when only one of the 8 strings tore, the Tzitzis is always valid, as the other side of that string is still intact, and hence there still remain 4 long strings. [Admur 12:2]

[10] Chesed Lealafim 12:1; Ashel Avraham Butchach 12 in name of Rav Menachem Mendel of Rimnov; Rebbe Rayatz in Sefer Hasichos 1944:56; Rebbe in Igros Kodesh 13:29 [printed in Shulchan Menachem 1:41]; See Admur 15:5 that implies such a Chumra does exist; Makor Chaim 12:15; Sfas Emes Shabbos 118b; Kaf Hachaim 12:12; Shulchan Hatahor 15:1; Piskeiy Teshuvos 12 footnote 2

Background:

It is a clear ruling in Gemara and Poskim, as Halacha of Moshe from Sinai, that Gardumei Tzitzis are Kosher. It is not recorded in any previous Poskim that there is an act of piety involved in nevertheless not using such Tzitzis. Nonetheless, being that the above Tzaddikim do make mention of this Hiddur, therefore one is to abide by it and fulfill the Mitzvah of Emunas Chachamim. [Ashel Avraham Butchach ibid; See there that he went ahead and fixed his Tzitzis that had one torn string after he heard of this Hiddur from Rav M.M. of Rimnov.]

[11] The reason: As the 32 strings have Kabalistic meaning and hence one should always be careful that it retain this number. [Chesed Lealafim ibid] Having the 32 strings protects one from stumbling in matters of promiscuity [Piskeiy Teshuvos ibid in name of Sefarim]

If the string is still 4 cm long: If the torn string is 4 cm long, then some write it is not necessary from even a standpoint of piety to replace the Tzitzis, although one who in any event desires to do so out of Hiddur Mitzvah may do so. [Ashel Avraham ibid]

The prohibition of removing Kosher Tzitzis from a Tallis: Although it is forbidden to remove Kosher Tzitzis from a Tallis, nevertheless, it is permitted to do so if one of the strings have torn and one plans on placing another pair of Tzitzis on the corner, in order so one have 8 complete strings. This applies even if the tear did not invalidate the Tzitzis. [Admur 15:5] See Ashel Avraham ibid that questions this allowance, stating that perhaps it is forbidden to transgress the prohibition of removing Kosher Tzitzis for a matter that is not brought in Poskim, nevertheless he concludes that one may do so, as it is nevertheless a Hiddur. See however Shulchan Hatahor ibid that writes one is to reuse these same Tzitzis by tying another string to the torn area. Other Poskim are also stringent in this matter. [Mor Uketzia 15; Artzos Hachaim 15:2]