One is allowed to untie tzitsis from one shawl and to place them on another shawl; however, he can not untie them without transferring them to another garment. Rema: This [general] prohibition of untying tzitzis applies specifically to one who is obligated to wear these garments, however it is permitted to untie the tzitsis of a dead person [without seeking to place them on another garment.] (See Mordechai and Tosafot Shabbat 22a).
One is not able to take the corner [of a garment] with the tzisis and to sew it onto another garment because of the fact that we require it [tzitsyot] to be on the "corner of their garments " (Numbers 15:38), and this corner [that is being sewed on] was not on the garment when it was made.
A tallit, that tzitzit was placed on it in accordance with the law, that was split into two, and in each part there is enough measurement [of garment] to wrap oneself with, and there remained on each one of them 1 or 2 [corners] of tzitzit, there is no concern for [the invalidation of] you should make it and it should not be made.
If the tallit became torn within 3 fingerbreadths near the edge of the corner, one is not permitted to sew it back up. And the explanation of Rashi is that we're worried that there will be left over from the sewing thread, and he will leave it there [hanging off of the corner] and he will add on to it 7 more strings for the sake of tzitzit. And according to this reasoning, even if [the corner] was torn any amount one should not sew it up. And according to this, a tallit of wool that became torn within 3 [fingerbreadths] it is permitted to sew it up nowadays, since it's not the normal way to sew with a wool string. And Rav Amram explains that the reason [one should not sew up the tallit] is because a tear within 3 [fingerbreadths] it [the torn corner] does not come into the realm of having the status of a garment, and it's compared to [a garment] that has no [measurement], and even if was sewed back on, it is considered that it was already set [in its status] and if one were to make tzitzit for it, it would not exempt the tallit. And according to this explanation, if [when the corner] was torn and there remained any part at all [attached to the tallit] it is valid. And there are those that sasy that according to Rav Amram he only invalidates the tzitzit that were on it at the time that he sewed it back on, but if after he sewed it he placed on it tzitzit it is valid. And a G-d fearing person complies with all of [the opinions] where it's possible.
If it tore downward from the hole that the tzitzit hang from, if the tzitzit were put on [the garment] before the tear, so that those tzitzit were there at the time of the tear, it is valid. And if it tore and there remained a little bit [of the garment untorn] and he sewed it up, and afterwards he put tzitzit on it, if it (the tzitzit) is wool then it is valid according to all opinions, and if it is of other materials that it's the normal way to sew with threads of those materials, he should not sew it according to Rashi. And if it tore and he sewed it up and afterwards he put tzitzit on it, it is questionable [if that is valid].
One who sews a piece of cloth on the corners of a tallit, and so too that which we are accustomed to sew around the hole that the tzitzit are in, if the tallit is made from silk and he sews with a thread of white silk, there should be caution with this matter according to the opinion of Rashi that there shouldn't be any sewing below 3 [fingerbreadths] and above the [measurement of] the knuckle of the thumb [from the corner]. Rema: And so too is the law in any place that one sews with a thread that is the same material as the tzitzit since we're worried that maybe he'll take that thread [to be used] for the strings of the tzitzit (T"H chaper 46 in the name of the Haghos Maymoni).