Gemara

102b (Mishnah): [On Shabbos] one may tie a string [of a harp that broke] in the Mikdash, but not outside the Mikdash.

Contradiction (Beraisa): If a harp string broke, one may not make a knot, but one may make a bow.

Resolution: Our Mishnah is R. Yehudah [who says that making a bow is a Melachah], and the Beraisa is Chachamim.

105a: The first Tana permits tying a string of a Levi's harp that broke;

R. Shimon (103a) permits only a bow. He explains that this is because it is only Shevus. It cannot lead to a Chiyuv Chatas. One may not tie a knot, which could lead to a Chiyuv Chatas.

Menachos 27a (Beraisa): One fulfills the Mitzvah of the four species even if they (the Lulav, Hadasim and Aravos) are not tied together;

R. Yehudah says, they are Kosher only if they are tied together.

(Beraisa): It is a Mitzvah to tie the Lulav (with the Hadasim and Aravos). If it is not tied, it is Kosher.

This is like Chachamim. It is a Mitzvah to tie it to beautify the Mitzvah - "Zeh Keli v'Anvehu."

Sukah 33b (Beraisa): If it became untied on Yom Tov, he ties it like a vegetable bundle.

Question: He should make a bow!

Answer: The Beraisa is R. Yehudah, who says that a bow is a proper knot.

Question: R. Yehudah requires a proper Eged (tying)!

Answer: The Tana holds like R. Yehudah regarding a bow, but not regarding Lulav Tzarich Eged (the Chiyuv to tie the three Minim together).

Rishonim

Rif and Rosh (Sukah 16a and 3:12): The Halachah follows Chachamim. It is a Mitzvah to tie only for beauty. Therefore, if one did not put it in the bundle before Yom Tov, he may put it in on Yom Tov. On Yom Tov, he ties it with (Rosh -- like) a vegetable bundle. He twines two ends of the Lulav leaves, and inserts them in back of the bundle. Alternatively, he makes a bow.

Ran (DH v'Hashta): Since we hold that Lulav Ein Tzarich Eged, if one did not put it in the Hadasim before Yom Tov, he puts them in on Yom Tov. The Rif says that if it required Eged, it would be forbidden due to Melachah on Yom Tov, since it is needed to be Machshir it, just like one may not diminish berries on a Hadas [on Yom Tov] to be Machshir it. Since we hold that Lulav Ein Tzarich Eged, he does nothing. He would be Yotzei even if he took each one by itself!

Ran (DH Lo): A vegetable bundle is edible, so it is not Muktzeh.

Ran (DH Iy): The Halachah does not follow R. Yehudah, who says that making a bow is tying.

Hagahos Ashri: The Rosh connotes that when tying it from Erev Yom Tov, it is a proper knot, and not like a vegetable bundle, which is not a proper bundle.

Rambam (Hilchos Lulav 7:12): One may tie the Lulav with a string or any species he desires, since tying is not Me'akev.

Poskim

Shulchan Aruch (OC 651:1): It is a Mitzvah to tie the four Minim through an absolute knot, i.e. a double knot, for beauty.

Beis Yosef (DH Kasav): The Mordechai (Sukah 748) says that the Re'em holds that one must tie two knots, one on top of other, for a Beraisa says that if it became untied on Yom Tov, he ties it like a vegetable bundle. This implies that during the week one must tie it in a way forbidden on Shabbos. Only a double knot is forbidden.

Gra (DH b'Kesher and DH Yesh): On Shabbos we require two knots [to be liable]. On 74b, we say that one is liable for sewing two stitches only if he tied them. If not, they will come apart. If so, every case of sewing is already liable for tying! Rather, he ties a single knot (one is not liable for this for tying), and through the two stitches or taking both ends out on one side, a single knot endures like a double knot. Rashi says so. The Rosh, Mordechai and other Poskim disagree, and explain that there are two knots.

Kaf ha'Chayim (9): Since it is a Mitzvah to tie, for beauty, one should say beforehand l'Shem Yichud...

Mishnah Berurah (8): Agurah b'Ohalecha says that the custom to insert the Lulav into a holder made of Lulav leaves (Kushiklach) without properly tying it is also considered tying.

Kaf ha'Chayim (10): If one wants to use a holder, it is proper to tie them first, and then insert them in the holder to be Yotzei according to all opinions.

Kaf ha'Chayim (11): It is proper to tie them so the Aravah leaves will not spread in different directions (Shlah). This means that they will touch the Lulav. The same applies to the Hadas leaves.

Shulchan Aruch (ibid.): The same species is not a Hefsek (interruption), but a different species is a Hefsek

Gra (DH v'Yashpil): Even though taking through something else is called taking, l'Chatchilah, when possible, we do not do so.

Shulchan Aruch (ibid.): If one did not tie it before Yom Tov, or it became untied, one cannot bind with a proper knot on Yom Tov. Rather, he binds through a bow.

Rema: Some say to bind differently, to wrap around the three Minim, and wrap the end of the tie in the wrapping.

Beis Yosef (DH Kasav): Avi ha'Ezri says that our custom is to tie it once, and then to wrap it around to make it tight as if he tied the two ends twice.

Beis Yosef (DH u'Mah she'Chasav v'Im): Rashi (33b DH Hutar) explains that the Gemara suggested that he should merely wrap the tie around the bundle and stick the end into the wrapping, but not tie the two ends together, like other knots. Making a permanent knot is an Av Melachah. This is permanent, for he never plans to untie it! It seems that Rashi's text said k'Agudah, like the Rosh's text. The Ran connotes that the Rif's text said b'Agudah. Since we hold unlike R. Yehudah [who holds that a bow is a proper knot], one may tie it with a bow. The Rif and Rosh say so. I say that also a string is not Muktzeh, for one can use it to make a bow.

Magen Avraham (3): This is the custom even on weekdays.

Mishnah Berurah (11): This refers to Yom Tov. The Shulchan Aruch gave the Heter of a bow, and the Rema gives the Heter of wrapping. I do not understand the Magen Avraham. The Mordechai mentioned the Heter of wrapping even during the week, but not like this! Rather, he makes one knot, and then wraps, and this is considered like two knots. Without a knot underneath, it is no better than a bow! Also the Gra questioned this. Therefore, I say that the bottom of the three knots [mentioned below], which ties the three Minim together, should have a knot first.

Rema: One inserts the Hadasim higher than the Aravos.

Magen Avraham (4): This is based on Kabalah.

Gra (DH v'Yesh): It is because the Medrash says that the Hadasim are like eyes, and the Aravos [are like lips].

Mishnah Berurah (12): The Hadasim should be on the right even for a left handed person.

Kaf ha'Chayim (14): The Ari Zal says that one Hadas is on the right, one on the left, and one is in the middle.

Rema: Some say to make three knots on the Lulav. This is the custom.

Taz (1): There is no source in the Gemara for knots on the Lulav itself. Even R. Yehudah requires only Eged of the three Minim together, and Rabanan say so for beauty. The Mordechai brings from Avi ha'Ezri '"Kapos [Temarim]" is written without a Vov, like Kapas (singular). For Vov (six) days it is tied, like one Kaf, and on the seventh day we untie it.'

Note: Perhaps we read it like 'Kafus' (bound). Had the Torah written it with a Vov, we would require two Lulavim.

Taz (1): I do not understand. One must rustle the leaves [while shaking it]. The knot prevents this! Therefore, one must do it in a way that enables rustling well. In any case, the lowest knot, which ties the three Minim together, counts towards the three knots on the Lulav. One makes two more above, if possible.

Magen Avraham (5): The three knots correspond to the three Avos.

Mishnah Berurah (14): Eliyahu Rabah connotes that there are three knots on the Lulav, excluding what ties the three Minim together. The top Tefach should be without a knot, to enable rustling the leaves well.

Sha'arei Teshuvah (3): Maharil says that for six days only the top Tefach is open, and on Hoshana Raba we untie until the middle of the Lulav. Eshel Avraham (3) said that if one did not have detached Lulav leaves to tie it, he may detach from the Lulav, since it is already detached; see 321:13. (One may remove honey from a hive only if it was scraped off the walls before Shabbos.) In 322:6, he seemed unsure. He said that we cannot learn from there (the Heter to roll fragrant branches and detach leaves), since they are proper to smell. People are stringent not to detach on Yom Tov. One may not permit this to them. If on Yom Tov one has no other way to tie, he detaches [leaves] with his teeth, preferably in private. We permit for a Mitzvah through a Shinuy.

Kaf ha'Chayim (16): The Ari Zal says to make 18 knots on the Lulav. Some say that this is with the three of the bundle, and some disagree.