Gemara

(Rava): The Torah mentions Yetzi'as Mitzrayim regarding Tzitzis to teach that just like Hash-m knew the true fathers to kill the firstborns, He knows who puts Kala Ilan (blue tree sap) on his garment and says 'it is Techeles.'

Menachos 38a (Mishnah): Techeles is not Me'akev the Lavan (threads that are white, like the garment), and Lavan is not Me'akev Techeles.

(Gemara) Suggestion: Our Mishnah is unlike Rebbi!

(Beraisa - Rebbi): "U'Re'isim Oso" - Techeles and Lavan are Me'akev each other;

Chachamim say, they are not Me'akev each other.

Rejection (Shmuel): The Mishnah is even like Rebbi. It teaches that the order is not Me'akev.

(Beraisa): The Mitzvah is to put the Lavan first. If one put Techeles first, he was Yotzei, but he missed (ideal fulfillment of) a Mitzvah.

(Rava): 'Lavan is not Me'akev Techeles' teaches that if the Lavan snapped and the Techeles remains, it is Kosher.

40a (Beraisa - R. Eliezer bar R. Tzadok): Chachamim exempted a linen garment from Tzitzis due to ignoramuses. They did not teach that only the Mitzvah of Techeles permits Kilayim, lest they wear Tzitzis dyed with Kala Ilan.

Question: Even if they do so, even though they do not fulfill the Mitzvah of Techeles, they fulfill the Mitzvah of Lavan!

Answer: Since the Mitzvah of Lavan can be fulfilled with linen threads (without transgressing Kilayim), the Mitzvah does not override Kilayim.

42b (Rav Shmuel bar Rav Yehudah): To dye Techeles, we take the blood of the Chilazon (a fish) and cook it with dying ingredients such as alum...

44a (Beraisa): The Chilazon has the form of a fish. Its color is like the sea. Techeiles is made from its blood. The Chilazon comes up only once every 70 years, therefore Techeiles is expensive.

Megilah 6a: Zevulun was complaining about his portion. Hash-m said 'I will make your brothers need you, through the Chilazon.'

Rishonim

Rambam (Hilchos Tzitzis 2:2): To dye Techeles threads, we take the blood of the Chilazon, a Techeles-colored fish whose blood is as black as ink. It is found in Yam ha'Melach. We cook its blood with cumin and other ingredients like dyers do. We put the wool in until it is like the color of the sky.

Question (Tiferes Yisrael, introdction to Mo'ed, Klalei Bigdei Kodesh DH Hinei): Yam ha'Melach is in Yehudah's portion. The Chilazon is found in Zevulun's portion!

Answer (Ma'adanei Yom Tov, on Halachah 7 of Rosh Halachos Kilayim (in Nidah Perek 9),10): The Rambam uses 'Yam ha'Melach' to refer to salt-water seas.

Rambam (4): If we checked Techeles and found out that it was dyed with another blackening agent that does not last, it is Pasul.

Ba'al ha'Ma'or (Shabbos 12a): I hold that Lavan and Techeles are Me'akev each other. Even though a Beraisa says that they are not Me'akev the other, we establish it to be like an individual, R. Yitzchak. Even though the Beraisa called his opinion 'Chachamim', the Halachah follows Rebbi against an individual. Amora'im strove to explain the Mishnah like Rebbi, unlike the simple reading. This is why Bahag omitted this Mishnah. Our custom follows Bahag.

Rebuttal (Milchamos Hash-m): The Amora'im strove to explain the Mishnah like Rebbi, for he codified it! Gitin 4a says that we tried to explain the Mishnah like R. Meir, even though the Halachah does not follow him, for a Stam Mishnah is (usually) R. Meir. Here, we follow R. Yitzchak, for his opinion is called Chachamim. Also, R. Noson, R. Yosi ha'Glili, and R. Yochanan ben Nuri all hold like R. Yitzchak. We rule like Rebbi against a colleague, but not against two or more colleagues. Also, the Gemara said that even Kala Ilan fulfills the Mitzvah of Lavan (even without Techeles). According to the Ba'al ha'Ma'or, there is no Mitzvah nowadays. If he will say that Chachamim decreed, where did they decree? One may not wear Tzitzis that do not fulfill the Mitzvah in Reshus ha'Rabim on Shabbos! From the days of the Ge'onim, we wear Tzitzis with only Lavan. Chachamim always enact like Torah law. If both were Me'akev, Chachamim would not enact to wear just one, for this does not commemorate the Torah Mitzvah!

Tosfos (42b DH v'Simemanim): It is astounding that one may mix other ingredients with Techeles! Perhaps together with the ingredients it is called Techeles. Rashi explains that the Techeles is soaked in other ingredients.

Medrash Rabah (Shlach 17:5, cited by Gra OC 9:3): Nowadays, we have only Lavan, for Techeles was Nignaz (hidden).

Etz Yosef: This means that it was forgotten.

Poskim

Shulchan Aruch (OC 9:2): Nowadays that there is no Techeles, we do not put wool strings on a linen garment or vice-versa.

Mishnah Berurah (7): Shatnez in Tzitzis was permitted only when we had Techeles and totally fulfilled the Mitzvah.

Noda b'Yehudah (1 OC 1): The Shulchan Aruch (32:48) disqualifies Tefilin boxes plated with gold. This is when the covering stands by itself and is not Batel to the box. When we blacken Tefilin, the ink is Batel to the leather of the box. It is not a problem of adding to the Mitzvah or covering the boxes. We learn a Kal va'Chomer from Techeles. The Torah was very particular about it, and even so one may mix other ingredients with it. Tosfos was astounded by this. I say that since Techeles dyes the wool only through the ingredients, the ingredients are part of the dye.

Question (Igros Moshe YD 2:133 DH Aval): What was the Noda b'Yehudah's proof from Techeles? Tosfos asked that since the Torah requires Techeles, other ingredients may be added only if they are Batel to it. One may not be Mevatel l'Chatchilah! Tosfos answered that perhaps it is called Techeles only with the ingredients. I.e. the tradition was that the blood must be mixed with ingredients. If the blood alone is used, it is Pasul! The blood itself is black, and the other ingredients make it bluish. The Rambam disqualifies if another blackening agent was used. He did not mention Kala Ilan, for any other blackening agent disqualifies. Perhaps he knew of others that become like Techeles through other ingredients. All such agents other than Techeles do not last. Presumably, if wool was dyed Techeles, and afterwards it was dyed with Kala Ilan, they are disqualified. Surely, if Kala Ilan was mixed with other blackening agents, it is Pasul. The blood must be the only blackening agent.

Yeshu'os Malko (OC 2): It seems that Techeles was Nignaz in the days of the last Amora'im. Tana'im argue about whether or not Techeles is Me'akev the Lavan. The Halachah was not decided, and there was danger. The Ari Zal wrote that there is no Mitzvah of Tzitzis nowadays, for the Halachah was decided that it is not Me'akev. It says 'l'Dorosam" regarding Tzitzis, a guarantee that the Mitzvah will be for all generations. It does not say so about Techeles. The Ba'al ha'Ma'or and Ri bar Baruch rule like Rebbi, that nowadays Tzitzis is only mid'Rabanan. All the Poskim rejected them. Rav Yuda Lebush ha'Kohen said that the tint (ink) fish is the Chilazon, for it has all the Simanim. This is mere reasoning. It is found every year, and in other seas as well. He said that we are obligated to try to fulfill the Mitzvah, even if it is a Safek. The Gemara said that Kala Ilan does not fulfill the Mitzvah of Techeles, but it fulfills Lavan! Most Poskim hold like this; some disagree. However, since we do not know that clearly that it is Techeles, and Gedolim hold that nowadays it is Nignaz and we need not have Techeles, we should not abandon what is Vadai and be stringent due to Safek. I do not say to remove this (alleged) Techeles, since the Tzitzis are not disqualified.

Beis Yosef (8 DH v'Nikreis): Ba'al ha'Itur commends to make a commemoration of Techeiles (and separate two strings from the other two).

Aruch ha'Shulchan (2): Years ago, someone claimed that he found the Chilazon and made Techeiles. He attracted some followers; the Gedolim and Klal Yisrael did not agree. It is Batel until Mashi'ach will come.

Shulchan Aruch (5:1): Some say that the Tzitzis must be the same color as the garment. Meticulous people do so.

Rema: Ashkenazim make Tzitzis only white.

Taz (8): A Meticulous person should make a white garment and white Tzitzis to fulfill all of the opinions.

Eshel Avraham (Pri Megadim 6): Eliyahu Rabah (9:5) says that his grandfather made white Tzitzis for a blue garment, to commemorate Tzitzis. Our custom is like this. Nowadays, people make a blue brim on the garment. The Tzitzis are the color of the primary part of the garment. Teshuvos v'Hanhagos (4:5): Piskei Tosfos (Menachos 118) says that one who deviates is called a commoner, for he makes it as if there are two Toros in Yisrael. Perhaps he transgresses Lo Sisgodedu (making factions). Also, some say that one is Yotzei only if one or two complete strings are Techeiles. If one received this custom from his fathers or Rebbi he may persist, but he must stipulate that if it is not the Mitzvah, he does not intend for the Mitzvah, lest he transgress "Bal Tosif." Others should not begin to wear Techeiles.

Ohr l'Tziyon (2:44:13): If one's fathers did not wear Techeiles, he should not. If it is not really Techeiles, there is concern for Kala Ilan (pretending to fulfill the Mitzvah).