Gemara

"Sha'atnez" teaches that the Torah forbids only (if the wool and linen were) Shu'a (combed), Tavuy (spun) and Nuz (twined or woven).

Question (Rav Ashi): Perhaps the Torah forbids even if the wool and linen were Shu'a, Tavuy or Nuz!

Answer: The Torah taught all of these in one word, to teach that all are required.

Shabbos 74b (Mishnah): Sewing two stitches (is an Av Melachah).

Question: Two stitches will come apart! (Anything that will not last is not a Melachah.)

Answer (Rabah bar bar Chanah): He is liable only if he tied them.

Kil'ayim 9:10 (Mishnah): One Tekifah (stitch) is not connected. There is no problem of Kil'ayim. One who unties it on Shabbos is exempt. If one made both ends to one side, it is connected and Kil'ayim applies. One who unties it on Shabbos is liable;

R. Yehudah says, (it is not connected) until he does three stitches.

Rishonim

Rambam (Hilchos Kil'ayim 10:24): If a wool garment was connected to a linen garment through one stitch, is not connected and it is not Kil'ayim. If he gathered both ends of the thread like one, or sewed two stitches, it is Kil'ayim.

Rosh (Hilchos Kil'ai Begadim, b'Sof Nidah, 17): One Tekifah is when he inserted the needle once. This is not connected for Tum'ah or Taharah. If one garment became Tamei, the other remains Tahor. If he sprinkled (Mei Chatas) on one of them, the other does not become Tahor. One who unties it on Shabbos with intent to sew it again is liable. Since it is not connected, this is not like tearing with intent to sew. 'If both ends are on one side' means that he passed the needle once and did not pass the entire thread through, and inserted the needle again, so both ends of the threads are on the same side. He tied the two ends together. If not, it would not last, like it says in Shabbos (74b). If he inserted the needle once, even if he tied the two ends on the edge of the garment, it is not connected

Tosfos (61b DH Shu'a): Rashi explains that (Shu'a is like) She'i'a, the Aranme'ic translation of Chlalak, for we divide it together with a comb. It is Tavuy (spun) together and Nuz (woven). Chachamim decreed about what is woven, even if it is not Shu'a and Tavuy. This is difficult. If so, why was a verse needed to permit Sha'atnez in Tzitzis? The (wool) Techeiles threads are combed and spun by themselves! Also, if Nuz means woven, what was the question 'perhaps the Torah forbids even if the wool and linen were Shu'a, Tavuy or Nuz!'? How could Shu'a or Tavuy be Sha'atnez without Nuz, since weaving connects them!

Aruch l'Ner: Rashi holds like the Tur, that when they are tied together, Sha'atnez applies even if they are not Shu'a, Tavuy or Nuz. (Maharsham - also the Ra'avad on Toras Kohanim says so.) Tosfos' latter question is not difficult. Rashi said that Shu'a and Tavuy are together. This connects them!

Tosfos (ibid.): Also why must "Sha'atnez" teach about woven? "Yachdav" already teaches that two stitches attach them, and all the more so if they are woven together! R. Tam says that "Sha'atnez" teaches that it must be Shu'a, Tavuy and Nuz. Each is Shu'a by itself, and also Tavuy, and also Nuz. Afterwards, they are connected. Nuz means twined. We need a verse to permit Sha'atnez in Tzitzis, for the Torah specifies Pesil (this is like Pesaltol, i.e. twisted) Techeiles, i.e. Tavuy and twined. We learn that the same applies to the white strings.

Poskim

Shulchan Aruch (YD 300:2): If a wool garment was attached to a linen garment through one Tekifah (he inserted the needle once), this is not connected, so it is not Kil'ayim. If he gathered two ends of the string together, or inserted two Tekifos, this is Kil'ayim.

Bach (3): The Rambam explains that the Mishnah discusses without tying anything. Therefore, one is exempt for one stitch, since he did not tie the ends on the edge of the garment. If he inserted the needle a second time, it is connected even if he did not tie the ends together. The Mishnah does not discuss one who tied the ends over the edge of the garment. Then, obviously one is liable even for one insertion of the needle.

Taz (2): 'He gathered the two ends' means that he tied them together.

Taz (3): The Mechaber is like the Rambam. In Hilchos Shabbos (10:9) the Rambam obligates one who sews two stitches only if he ties the ends of the thread on each side, so that the stitch will endure and not come out. We must say that connection for Kil'ayim is different. Any amount is considered connected (even if it will come out), for the Torah said 'Yachdav". This teaches that the wool and linen may not have even a temporary connection. One stitch is not even a temporary connection. The Mishnah seems to equate Kil'ayim and Shabbos, but this is only for the difference between one stitch and two. Regarding tying on Shabbos, it must last, for the Torah forbade Meleches Machsheves.

Rema: Some say that it is Kil'ayim only if there are two stitches and he tied both ends of the thread.

Tur: The Rosh holds that two stitches are connected only if he inserted the needle twice, and tied the two ends together. The Rambam holds that it is even if he inserted the needle once, and tied the two ends on the edge of the garment.

Beis Yosef (DH Chiber): Also R. Shimshon on the Mishnah in Kil'ayim explained like the Rosh.

Bach (DH u'Mah): The Rosh proved from Shabbos that if he did not tie them, they will not last. How can the Rambam answer this?! It seems that the Rambam agrees that the Mishnah discusses one who tied the ends. However, he explains that one stitch is not connected when he tied each of the two ends by itself, each on its own side, so that the stitch will endure and not come undone. This tying is not connection. Rather, one must tie the two ends on the side of the garment. This is why the Rambam wrote (regarding one stitch) 'if he gathered both ends of the thread like one', but regarding two stitches, even if he tied each end by itself and did not tie them together, it is connected. The Rosh holds that even if he tied them together, one stitch is not connected, unless he did two stitches and tied the ends together.

Taz (4): The Tur holds that if he did one stitch and tied the ends on the edge of the garment, it is not connected. It seems that this is because when it is tied on the edge, it is ruined quickly when anything touches it, for the thread is very thin, and nothing protects it. Why did the Torah need to permit Kil'ayim in Tzitzis? The strings are tied outside (on the edge)! Rather, only when one ties two matters together, and the thread can snap easily when it is outside, this is not considered connected for Kil'ayim. Tzitzis is different, for it is merely connecting the strings to the garment. They are not easily uprooted. The Rema did not distinguish based on how (the ends of) the thread is (are) tied. It seems that he is stringent, and (forbids) even if it is tied on the edge. This is proper, for the Rambam forbids even without tying them at all, and all the more so if one made a knot at the end of the thread on each side. The Rambam obligates for that even for Shabbos.

Gra (5): (In the Mishneh Torah, the Rambam) explains that even one stitch, if he also tied the ends on the edge of the garment, is connected, and all the more so two stitches. In Perush ha'Mishnayos, he explained that it is connected only if he made two stitches. The Rosh and R. Shimshon explain like this, and they also require tying the ends. The Rema brings their opinion. The Rambam holds that this is only for Shabbos.

Gra (Likut): The Yerushalmi says that one is liable for two stitches for Shabbos only if he tied the ends on both sides. The same applies to Kil'ayim. When there are many stitches, it last even without being tied.

Pischei Teshuvah (2): The Magen Avraham (11:13) says that even the opinion that one stitch is not connected even if it is tied says so only regarding a single knot. If one ties two knots, all agree that even one stitch is connected. If not, we would not find Kil'ayim regarding Tzitzis. Chavos Ya'ir (143) wrote similarly, and added that even though he holds that two stitches require a knot, three stitches, and all the more so more than three, do not require a knot.