Halakhah for Menachot 4:1
הַתְּכֵלֶת אֵינָהּ מְעַכֶּבֶת אֶת הַלָּבָן, וְהַלָּבָן אֵינוֹ מְעַכֵּב אֶת הַתְּכֵלֶת. תְּפִלָּה שֶׁל יָד אֵינָהּ מְעַכֶּבֶת שֶׁל רֹאשׁ, וְשֶׁל ראשׁ אֵינָהּ מְעַכֶּבֶת שֶׁל יָד. הַסֹּלֶת וְהַשֶּׁמֶן אֵינָם מְעַכְּבִין אֶת הַיַּיִן, וְלֹא הַיַּיִן מְעַכְּבָן. הַמַּתָּנוֹת שֶׁעַל מִזְבֵּחַ הַחִיצוֹן אֵינָן מְעַכְּבוֹת זוֹ אֶת זוֹ:
The [absence of the] blue [in the fringes] does not invalidate the white, neither does the [absence of the] white invalidate the blue. The [absence of the] hand-<i>tefillin</i> does not invalidate the head-<i>tefillin</i>, neither does the [absence of the] head-<i>tefillin</i> invalidate the hand-<i>tefillin</i>. The [absence of the] fine flour and the oil does not invalidate the wine, neither does the [absence of the] wine invalidate them. The [absence of one of the] sprinklings [of the blood] on the outer altar does not invalidate the rest.
Gray Matter II
The Tiferet Yisrael (Kupat Harochlim, Klalei Bigdei Kodesh Shel Kehunah; printed as an introduction to Seder Mo’eid) goes even further, arguing that even tzitzit do not actually require the chilazon’s “blood,” but rather can be made of any permanent (non-fading) blue dye (also see Drishat Tzion, Ma’amar Kadishin 3). He explains that the Gemara often contrasts the chilazon with a plant dye called kaleh ha’ilan because kaleh ha’ilan is the only dye that cannot be used for techeilet. Most authorities do not appear to accept the views of the Mirkevet Hamishneh and Tiferet Yisrael (see Mishneh Lamelech, Hilchot Klei Hamikdash 8:11, and Mishkan Shiloh p. 407). Indeed, Rav Yosef Dov Soloveitchik told me he believes that the bigdei kehunah are invalid without proper techeilet. which we have not used for centuries (see Rambam’s commentary to the Mishnah, Menachot 4:1). Great efforts have been made in recent years, however, to identify the chilazon as the Murex Trunculus snail (see Techumin 9:423-446). Dye from this snail has now been made available for use in tzitzit and could theoretically be used to dye bigdei kehunah. Although some prominent rabbis (such as Rav Hershel Schachter) treat many of the arguments for the use of the Murex Trunculus seriously, only time will tell if the observant community will widely accept this dye as authentic techeilet.14See Tekhelet: The Renaissance of a Mitzvah for essays by several Rashei Yeshiva of Yeshiva University regarding the use of techeilet from the Murex Trunculus in tzitzit. Rav Schachter, in his essay, describes the Murex Trunculus as safeik techeilet (possible techeilet). Also see Kovetz Teshuvot 2, where Rav Yosef Shalom Eliashiv rejects the use of the new techeilet.