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On Shabbat, there are certain types of knots that one may tie or untie and some that one may not tie or untie. See below for details and practical applications.
The Basics of Koshair
Where was it in the Mishkan? The gemara explains that Koshair was performed in the construction of the Mishkan by the trappers of chilazon who would tie knots in their nets in order to adjust their nets.
Definition: There is an important machlokes Rishonim as to the definition of a knot that would warrant a chiyuv min hatorah. According to Rashi and Rosh, as long as the knot is intended to stay tied forever or for a long time, one is chayav min hatorah for tying it. According to Rambam and Rif, not only must one intend to keep it tied for a long time to be chayav min hatorah, it must also be defined as a kesher uman - professional knot, such as a sailor’s or camel-driver’s knot.
Permanence:There are different opinions as to what constitutes a ‘long time.’ The Kol Bo says that a knot that is tied for more than one day is called somewhat permanent, and it is rabbinically forbidden to tie. The Tur and Mordechai say that a knot that is tied for more than seven days is permanent. Rabbeinu Yerucham says that only a knot that is tied for half a year or a year is considered permanent.
- The Melacha of Koshair is the binding two pliable items by intertwining them such as through a firm knot or twining threads into rope. 
- A knot that’s forbidden on a Torah level is one that is both (1) professional and (2) is meant to be permanent. 
- Many Rishonim hold that it is rabbinically forbidden to tie a knot which is either professional or meant to last for more than 24 hours. Some Sephardim hold that a non-professional knot is only rabbinically forbidden if it meant to last a week. Some Rishonim hold that a knot which is permanent even if it's not professional is Biblical prohibited.
- A knot which is both non-professional and meant to be undone within 24 hours may be tied.  Some Sephardic authorities permit a non-professional knot which is meant to last less than 7 days. 
- Even if you personally intend to leave it less than 24 hours, if it’s a knot sometimes left permanently by people it’s forbidden Rabbinically to prevent confusion between temporary and permanent knots. 
- Any knot which is forbidden to tie is also forbidden to tighten. 
- For example, one may not tighten the double knot of the tzitzit or talit if it became loose. 
- The rules of untying are parallel to the rules for tying. Thus, any not that is permitted to tie is also permitted to untie.
- According to some opinions, matir only applies if you are untying with intention to re-tie a better knot. Others disagree.
- If someone attempted to tie a slipknot but then they made it into a real knot by accident, they may untie the knot because the intent from the beginning was to untie it. 
- A bow-knot, slipknot, or a single-knot isn’t a true Kesher-bind and is permissible even if it does remain in place for a while.  For example: a ribbon on a gift, a single bind on a shopping bag or garbage liner.
- It is permitted to tie a slip knot or bow without a underlying knot even if it is tight and meant never to be undone. 
- After tying a bow knot (a bow on top of a single knot), as is common when tying one's shoes, one may not tie a second bow on top of the bow knot.  Others permit tying this knot since it can still be untied with a single hand by pulling on the end of the string. 
A bow on top of a single knot
- According to Ashkenazim, it is permissible to tie one's shoelaces with a bow on top of a single knot (initial winding) only if it is meant to be untied within 24 hours. According to Sephardim, it is permissible as long as it is meant to be untied within 7 days.  However, if one doesn't usually untie it within 24 hours (and manages to slip one's foot in and out) it would be forbidden to tie one's shoes on Shabbat. According to Sephardim it is permissible even if it isn't meant to be untied within 24 hours, yet it's preferable to make sure to untie it within 7 days. 
- Someone doing the wrapping of an Ashkenaz Sefer Torah (Gelilah) and has to tie a sash around the Torah, it's preferable to just wrap the sash around the Torah and tuck in the end, however, if that's not possible then some are lenient to tie a bow on top of a single knot  while others are strict unless one is planning on untying it within 24 hours. 
- Many authorities hold that it is permissible to tie a necktie on Shabbat if one usually unties it within 24 hours. However, if one doesn't usually untie it within 24 hours, some say that if one intends to untie it within 24 hours it is permitted, while others write that one should tie it from before Shabbat and put it on and take it off by just tightening and loosening the side which goes around the neck, however, one shouldn't untie it on Shabbat. 
- It is forbidden to tie a single knot with one string onto itself. 
- It is permitted to tie a single knot in two strands of string even if it is tight and meant never to be undone. 
- It is permitted to tie a single knot and then tuck in the ends of the string so that it doesn’t come out only if it is menat to be undone within 24 hours. 
- One may not add a second knot on top of a preexisting single knot, nor may one add a third knot onto a double knot in order to tighten it. 
- According to Ashkenazim, a tight double knot is considered like a professional knot and may not be tied. The Sephardic custom, however, is to be lenient if one intends to untie it within 7 days, yet, one who is strict will be blessed. 
- It is permitted to tie a loose double knot if it’s usually not meant to last 24 hours and this time specifically it's also not meant to last 24 hours.  For example, it’s permissible to tie a double knot for a belt (gartel) or sash (tichel) if it’s going to be untied in 24 hours since it’s not meant to be tight. 
- If unintentionally one pulled the bow-knot of the shoelaces and it became a double knot, one may untie it. 
- One can untie a double knot that is made on a candy bag given out at shul for a simcha. 
- One may tie a bandage on a wound with a loose double knot if it is meant to be untied within 24 hours. If it is meant to last for more than 24 hours but less than 7 days, one should use a adhesive bandage or a bow knot, however, if that isn't possible then it's permissible. 
- One may tie a head covering with a loose double knot as long as one usually unties it everyday. However, if one usually removes it within untying the knot then it's forbidden to tie it even if this time one's intent is to untie within 24 hours. 
Wrapping a string
- It is permitted to wrap a string around an object and then tuck the ends of the string in even if it is meant to be left permanently. 
Twist ties and Cable ties
- Some authorities forbid using wire twist ties to close bags unless one plans to undo it within 24 hours.  However, others permit.  It is praiseworthy to avoid using them if they will remain undone for more than 24 hours. 
- Some poskim permit using plastic cables ties since it is not a knot, but simply tightening. 
- Socks or gloves that are tied together should be detached before Shabbat, but if one forgot one should untie it but ripe it so as to destroy the string. 
- One can take out the pins put into shirts in packaging to keep it from wrinkling. 
- It’s totally permitted to thread new laces into shoes to replace old laces but it’s not permitted if the shoe was never laced. 
- Shabbat 74b
- Shabbos 112a s.v. b’dirabanan
- Shabbos 15:1
- Shabbos 10:1
- Shabbos 41b
- Cited in Rama 317:1
- As cited in Rema 317:1. Tur in 317 says that a knot is only permanent enough to be chayav chatas if it is able to be kept forever, but a knot meant to last seven days is patur aval assur.
- Cited in Beit Yosef 317:1
- Mishna Brurah 317:34 quoting the Rambam, Shemirat Shabbat KeHilchata 51:52
- Shulchan Aruch (317:1) in the opinion of the Rambam and Rif
- The Rama 317:1 quoting Tur and Mordechai; Mishna Brurah (317:6) explains it as anything less than 24 hours.
- Rama also quotes Kol Bo and Hagahot Maymoniyot. Chazon Ovadia Shabbat Part 5 page 47 and Yalkut Yosef (317:6) writes that Sephardim hold like this opinion.
- Magan Avraham (317:20), Kol Sinai (Koshair #3)
- Shemirat Shabbat KeHilchata 15:50 (in new editions 15:53), Kesot HaShulchan (Siman 123, at the end of note 4), Sh"t Az Nidabru 3:22, Sh"t Bear Moshe 6:3
- 39 Melachos (Rabbi Ribiat, vol 3, pg 799)
- Mishna 111b, Rambam Shabbos 10:7, Rama 317:1, Mishna Brura 317:7, Chazon Ovadia (Shabbat vol. 5, p. 47).
- Tosfos Shabbos 73a “Hakoshair”, Biur Halacha 317:1 “‘Dino’”. See 39 Melochos v. 3, p. 786, fn. 2.
- Rashi (74b “Shari”), see 39 Melochos, pg. 805.
- Sh"t Rivivot Ephraim 1:222:17
- Shulchan Aruch (317:5) and implied by Rama (317:1). Kitzur Shulchan Aruch 80:45 and Yalkut Yosef (317:11) agree.
- Shulchan Aruch (317:5), Beiur Halacha 317:5, Yalkut Yosef (317:11, Shabbat, vol 2, pg 562), The Shabbos Home (Rabbi Simcha Cohen, vol 1, chap 13, pg 207)
- 39 Melachos (Rabbi Ribiat, vol 3, pg 798)
- Orchot Shabbat (10:15) and footnote 28
- 39 Melachos (Rabbi Ribiat, vol 3, pg 798)
- Yalkut Yosef (317:11), Rav Ovadyah in Kol Sinai (Koshair #5). See previous two footnotes.
- Sh"t Tzitz Eliezer 7:29, Shalmat Chaim 185, Beir Moshe 6:55.
- Shemirat Shabbat KeHilchata 15:53 (in new editions 15:56) is strict unless one is planning to untie it within 24 hours, however, in the footnote (15:188) he quotes Rav Shlomo Zalman who said that the minhag is to be lenient.
- Rama (317:1) quoting the Hagahot Alfasi, Kitzur Shulchan Aruch 80:45, Shemirat Shabbat KeHilchata (in the new edition 15:52)
- Shemirat Shabbat KeHilchata (in the new edition 15:53)
- S”A HaRav 317:1, Yalkut Yosef (317:11, Shabbat, vol 2, pg 562), The Shabbos Home (Rabbi Simcha Cohen, vol 1, chap 13, pg 207)
- The Shabbos Home (Rabbi Simcha Cohen, vol 1, chap 13, pg 207-8)
- Shemirat Shabbat KeHilchata (in the new edition 15:53), Chazon Ovadia Shabbat Part 5 page 68 also says it's worthwhile to be stringent.
- Shemirat Shabbat KeHilchata (in the new edition 15:53), 39 Melachos (Rabbi Ribiat, vol 3, pg 799).
- Shemirat Shabbat KeHilchata 15:52 (in new edition 15:55), The Shabbos Home (Rabbi Simcha Cohen, vol 1, chap 13, pg 208)
- Shaarei Teshuva (317:1), Tzitz Eliezer 16:10, Menuchat Ahava 14:5, 39 Melachos (Rabbi Ribiat, vol 3, pg 793)
- Mishnat Halachot 3:38, Rav Ovadyah in Kol Sinai (Koshair 3:5), Shemirat Shabbat KeHilchata 15:56 (in new editions 15:60)
- Toldot Yitzchak O”C 2:11
- Shemirat Shabbat KeHilchata (in new editions 15:55)
- Shemirat Shabbat KeHilchata (in new editions 15:55)
- Rama 651:1, The Shabbos Home (Rabbi Simcha Cohen, vol 1, chap 13, pg 208)
- Shemirat Shabbat KeHilchata 14:9 and (15 note 166, in the new edition note 174) quoting Rav Shlomo Zalman
- 39 Melachos (Rabbi Ribiat, Melechat Koshair note 54) quoting Rav Moshe Feinstein
- 39 Melachos (Rabbi Ribiat, vol 3, pg 800)
- Chazon Ovadia Shabbat Part 5 page 75
- Shemirat Shabbat Kahilchata 15:63 in name of Rav Shlomo Zalman Aurbach, Az Nidabru 1:58.
- Shemirat Shabbat Kahilchata 15:64 in name of Rav Shlomo Zalman Aurbach, Bier Moshe 1:22.
- Rav Ovadya Yosef in the Haskama of Netiv Mitzvot, Mayim Chaim Meshash O”C 1:147, Bier Moshe 2:20(2),6:59, Tefilah LeMoshe 1:58.