- The pesukim in Shalach  introduce the mitzvah of Tzitzit and among them is one that mentions the requirement to have a string of the Techelet color.
"דבר אל בני ישראל ואמרת אלהם ועשו להם ציצת על כנפי בגדיהם לדרתם ונתנו על ציצת הכנף פתיל תכלת"
- The Mishna (menachot 38a) writes clearly that wearing Tzitzit without Techelet also fulfills the mitzvah of Tzitzit (while lacking a part of the mitzvah). This is codified in the Rambam (Tzitzit 1:3).
- From the end of the era of the sages of the Gemara Techelet became unavailable and extinct from common practice. However, in the last century and a half many have investigated this lost dye and some claim to have rediscovered it. The halachic question is how to deal with the dyes which some have claimed to be the real Techelet of the Torah.
- Rav Hershel Schachter in Ginat Egoz (Siman 2) writes that there is very little reason not to wear the Tzitzit because of a doubt that perhaps it is the correct Techelet. He addresses the fact that the minhag is to wear a white garment with white strings, but says clearly that there’s no reason that this minhag should override a possibility to fulfill a biblical mitzvah. Furthermore, the Bet HaLevi (Sh”t Bet HaLevi 42) who says that there’s a prohibition of detracting from the מצות by fulfilling a mitzvah in a inferior way but that since a positive mitzvah overrides a negative one, there’s still reason to wear Tzitzit without Techelet because one still fulfills a positive mitzvah. Extending this further, Rav Schachter writes that if a person owns Techelet then it becomes possible to fulfill the positive mitzvah and negative one simultaneously and therefore one wouldn’t be allowed to use a positive mitzvah to override a negative one. To see how Rav Schachter ties the Techelet Tzitzit see here www.tekhelet.com For a detailed how-to, see mywesternwall.net.
- There are other Poskim who hold of wearing Techelet but most don't hold it is as much as a Chiyuv as Rav Schachter. Rav Meir Mazuz, Rosh Yeshiva of Kiseh Rachamim in Bnei Brak, wears Techeles on Shabbos and Yom Tov as he holds it's not a Chiyuv rather a nice Chumra. 
- However, the overwhelming minhag for the last century was not to wear the Techelet which were suggested as the real species of Techelet. There’s a short article of Rav Elyashiv (Kovetz Teshuvot 1:2) who mentions several reasons to defend this practice. See Teshuvot vihanhagot 1:29 and 4:5 who quotes Rav Chaim Pinchas Scheinberg saying that future generations may think that we have had the tradition the entire time. Also, http://www.dailyhalacha.com/displayRead.asp?readID=785&txtSearch=techelet quotes Chacham Ben-tzion Abba Shaul who says that if your father and grandfather didn't wear a string of techelet than you should follow that practice as well. Rav Shlomo Aviner http://www.ravaviner.com/2011/06/techelet.html and in Shu"t She'elat Shlomo 4:12-20 also gives several reasons not to wear techelet. So does Rav Osher Weiss
- Firstly, the discoveries made regarding the Techelet have been inconclusive and the first two claims were later disproved and who’s to say that the new one which is claimed to be authentic will stand the test of time. Additionally, perhaps by the mere fact that for over a thousand years Jews haven’t used such a species for Techelet is a tradition that this species isn’t the correct one. Some sources even point to the fact that the secret of the Techelet was hidden after the destruction of the Bet Hamikdash. Lastly, how are we to determine the correct method in developing the dye if it’s disputed even in the early authorities and the dispute is left unresolved.
Number of Techelet Strings
The Radziner in Petil Techelet summarizes the dispute:
- Rambam holds that one string out of eighth are techelet. Sefer Hachinuch, Maggid Meisharim, Mabit, Maharam Ibn Gabay, Sefer Charedim, Levush, Gra, Sefer Hakaneh, and Arizal agree with the Rambam. Radziner in Petil Techelet ch. 6 ends up following the Rambam but says one has what to rely upon to follow either of the other opinions. Ben Ish Chai (Noach n. 17) and in Od Yosef Chay (Vaera n. 3) holds like the Rambam. Rav Yonatan Mazuz agrees.
- Raavad holds that one string out of four (two out of eight) is techelet. Aruch (Techelet) agrees with the Raavad. He cites the Sefer Haitur, Sefer Machriya, Tur, Rabbenu Yerucham, Shulchan Aruch Harav who agree. Smag has safek like Tosfot or Raavad.
- Rashi and Tosfot hold that four strings out of eighth are techelet. Rav Schachter ("Hilchos Tzitzis" beginning) holds like Rashi and Tosfot and recommends all Ashkenazim to follow this method due to the plethora of sources advocating it (Baal haIttur, Tur, Shulchan Aruch, etc.).
There are several methods of tying tzitzit with techelet.
The Rambam maintains that there should be between 7-13 links (chulyot) and that each link should have 3 winds (krichot). In the closest link to the garment, the first wind should be white, while the other two winds of that link should be techelet. In the farthest link from the garment, the final wind should be white, while the other two winds of that link should be techelet. All other links contain only techelet.
The Raavad holds that there should be 4 links and that each link has 7 winds. Each link is separated by a knot. The winds of each link should alternate white and techelet, the first and last wind of each link being white. The Raavad dictates that the link closest to the edge of the garment and the two links farthest from the garment must follow the alternating pattern. However, the unmentioned link can either be all white or the alternating white and techelet pattern.
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If the Strings Ripped
- If all white tzitzit, click here.
- According to the Rambam, if all white strings tore off up to the corner of the garment, and all that was left were techelet strings, the garment is still fit for use. 
- Bamidbar 15:38
- Kisei Rachamim's website
- Rambam Halachot Tzitzit 1:6
- Raavad on Rambam Halachot Tzitzit 1:6
- Rashi on Mishna of Menachot 38a
- Rambam, Halachot Tzitzit Chapter 1
- Raavad on Rambam Halachot Tzitzit 1:7
- Rambam, Hilchot Tzitzit 1:4. He explains that this is due to the mishna on Menachot 38a saying that the absence of white does not invalidate blue and the absence of blue does not invalidate white.