1) DIFFERENT TYPES OF KNOTS
OPINIONS: The Gemara explains that there are three types of knots with regard to the Melachah of tying: a knot that is forbidden to be tied and for which one who ties is Chayav; a knot that is forbidden to be tied but one is Patur for tying; a knot which is permitted to be tied. There are three different approaches in the Rishonim to explain these three categories of knots.
(a) RASHI maintains that one is Chayav for tying a knot to last for a long time, regardless of the quality of the knot (that is, whether it is a professional knot or an unprofessional one). One is Patur for tying a knot that is meant to last for an intermediate amount of time, although tying such a knot is still forbidden. One is permitted to tie a knot that is meant to last only for a short amount of time.
(b) The RE'EM cited by the MORDECHAI writes that if one has definite intention to leave the knot for a long time, he is Chayav. If his intention is that he might leave it for a long time, but he is not sure, then he is Patur. If he has definite intention to leave the knot only for a short time, he is permitted to tie it.
(c) The RIF and the RAMBAM rule that one is Chayav only when two criteria are met. First, the knot must be professionally tied, and it must be long lasting. If only one of these conditions are met, then one is exempt. If neither condition is met, then one is permitted to tie the knot. What defines a professionally-tied knot? The SHILTEI GIBORIM explains that it is a strong knot. The MISHNAH BERURAH adds that it is a knot that will never become undone by itself.
HALACHAH: The SHULCHAN ARUCH (OC 317:1) cites the opinion of the Rif and Rambam (c), that one is permitted to tie a knot only when it is not a professional knot and it is tied to last for a short time. The REMA cites the opinion of Rashi (a), that if it is a permanent knot, one is Chayav regardless of the expertise of the knot.
All of the abovementioned Rishonim (Rashi (a), Re'em (b), and the Rif and Rambam (c)) require that the knot be permanent in order for one to be Chayav for tying it. What is a permanent knot that one may not tie, and what is a temporary knot that one may tie?
From the words of Rashi and the Rif, it appears that a permanent knot is defined as a knot that one ties in order to leave it as a knot forever, with no intention to ever untie it.
A knot that is not permanent and that one may tie l'Chatchilah is defined by the REMA in the name of the TUR and MORDECHAI as a knot that one ties with intention to leave it for less than one week.
RABEINU YERUCHAM (cited by the BEIS YOSEF) writes that one may tie a knot as long as he intends for it to last for less than three days.
The Rema cites the KOL BO who rules like Rashi here, who says that a temporary knot that one may tie on Shabbos is one that is normally untied every day.
The Rema rules that, in practice, one should be stringent and not tie a knot on Shabbos with intention that it last for more than one day.
2) A SHOE THAT TORE ON SHABBOS
QUESTIONS: The Gemara says that Rebbi Yirmeyah was once walking in a Karmelis when one of the straps of his sandal tore. Rebbi Avahu permitted him to tie a temporary string around it so that he could wear it until he arrived home. The Gemara relates a second incident, in which Abaye's shoe tore while he was in a private courtyard, and Rav Yosef ruled that the broken shoe was Muktzah. The Gemara says the difference between the two cases is that in the first case, the shoe tore while Rebbi Yirmeyah was in an unprotected area, and in the second case, the shoe tore while Abaye was in a protected area.
(a) What difference does it make where the person was when his shoe tore? If it is Muktzah, it should be Muktzah regardless of where it tore!
(b) Why should the shoe be Muktzah altogether? Since it can be used for another useful purpose, it should be permitted to be handled.
ANSWER: The RASHBA cites the RA'AVAD who says that in the first case, Rebbi Yirmeyah was walking in a public area when the shoe tore. If he would have left the torn shoe there, someone else would have taken it to use for some other purpose. Thus, his broken shoe was considered to have be usable for some other purpose. In contrast, in the second case, Abaye's shoe tore in a private area (a Chatzer). Since no one else would have found the shoe there and taken it to use for another purpose, and the owner of the shoe had intention to mend the broken strap, the broken shoe indeed had no other permissible use on Shabbos. The owner had intention to use it only as a shoe, and he definitely will mend it after Shabbos to use as a shoe. Therefore, it is Muktzah.
(For this reason, an object such as a button that falls off of one's garment in a private area (such as one's home) and which one definitely will sewn back on after Shabbos should be considered Muktzah.)

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3) THE STATUS OF A REPAIRED SHOE
QUESTION: The Gemara cites the Mishnah in Kelim (26:4) which teaches that if one of the loops (for the strap) on the side of a sandal that was Tamei (with Tum'as Midras) tears and is repaired, the sandal remains Tamei with Tum'as Midras. If the second loop (on the other side of the sandal) tears and is repaired, the sandal is no longer Tamei with Tum'as Midras, because the sandal is unfit for use without loops for the strap. Nevertheless, it is Tamei with Tum'as Maga of Midras, because the repaired shoe "touched itself" (while it was Tamei with Tum'as Midras) before the second loop tore.
Why does the Mishnah in Kelim say that the loops were repaired after they tore? The Halachah should be the same even when the loops were not repaired! When the first loop tears, the sandal is still usable and thus it retains its Tum'as Midras, and, when the second loop tears, it is no longer usable as a shoe and therefore it loses its Tum'as Midras but remains Tamei with Tum'as Maga Midras!
ANSWERS:
(a) RASHI (DH Mai Lav) explains that it indeed does not make a difference whether the loops were repaired or not. Even if they were not repaired, after the first loop tears the sandal is still Tamei with Tum'as Midras, and after the second loop tears it is still Tamei with Tum'as Maga Midras. This is because even without its loops, the sandal is a usable utensil and can become Tamei with Tum'as Maga Midras. The reason why the Mishnah in Kelim says that the loop was repaired is because it wants to teach that even though the first loop was repaired and the sandal became a completely usable sandal, it still loses its Tum'as Midras when the second loop tears, because the sandal is not the same sandal that it was originally.
(b) TOSFOS (DH Sandal) points out that Rashi here (in DH Mai Lav) follows his original explanation, which is printed in the margin of Rashi in our texts. According to that explanation, even though the straps of the shoe are torn, the shoe is still considered to be a usable utensil because it can be used for some other useful purpose. However, in his final version, Rashi (DH Aval Tamei) writes that once both straps are torn, the shoe is not considered a utensil at all, and if both straps were to be torn at one time, the shoe would be completely Tahor. Therefore, the Mishnah must say that the first strap was repaired before the second strap broke, because, otherwise, once the second strap broke and both straps were broken at the same time, it would not be a utensil and would not become Tamei at all.
Tosfos proves from the Mishnah in Kelim (26:4) that if both straps were torn at the same time, the sandal would indeed be Tahor, like the second version in Rashi. However, Tosfos points out that the second strap that tore did not have to be repaired in order for the shoe to be Tamei with Tum'as Maga Midras. The only reason the Mishnah says that the second strap was repaired is to be consistent with the first part of the Mishnah, which necessarily states that the first strap was repaired (for if it was still ripped at the time that the second one ripped, the shoe would be entirely Tahor). (See also Insights to Eruvin 24:2.)

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