We will begin with a brief overview of the basic idea of the series of knots and spirals of the tzitzit (without getting into their deeper meaning) in order to help understand the answer to your question.
The Torah calls the string appendages to the corners of our garments “tzitzit.”[1] This word, which can also refer to hanging strands of human hair,[2] implies that they are hanging strings. The Torah elsewhere[3] speaks of them as “gedilim,” which implies that there is a string that is wrapped around other strings.[4] In addition, at least one knot is needed for each set of strings that is connected to a corner of the garment. This is derived from the latter source, which mentions tzitzit in direct proximity to the prohibition of sha’atnez.[5] This juxtaposition teaches us that under the right circumstances, the mitzva of tzitzit allows one to wear a linen garment with woolen tzitzit, since the positive mitzva of tzitzit overrides the negative commandment against sha’atnez.[6] Furthermore, the juxtaposition implies that the strings must be tied on to the garment, for if they were hanging loose, there would be no issue of sha’atnez.[7]
The practice is to have five double-knots alternating with four sets of wrappings, which have different numbers of revolutions of the wrapping string. However, these numbers are not Torah law. The gemara[8] states that one set of knots is required by the Torah; a simple single knot is not permanent enough to count halachically. The gemara[9] proposes a proof that there should be knots around each gedil, but then deflects the proof. There is no clear consensus among the Rishonim as to what the conclusion of the discussion is.[10] However, it appears accepted that neither the customary number of gedilim nor the number of knots in each set is critical.[11]
In fact, the Beit Yosef[12] says that if one does not have enough time to tie all the knots as prescribed, one section of wrappings and one set of knots is sufficient. His example of a pressured time is Erev Shabbat, in which case one will enter Shabbat with non-standard tzitzit. This suggestion is quite instructive given the halacha that a four-cornered garment in which not all the tzitzit are attached in a manner that fulfills the mitzva is considered to have useless appendages, and one is forbidden to wear it on Shabbat because of the prohibition of “carrying” if there is no eiruv.[13] Thus, we see from the Beit Yosef that even if each corner has only one halachically valid knot, the tzitzit are fundamentally valid, and one may wear those tzitzit without a problem of Shabbat. (When Shabbat concludes, one should rectify the situation promptly.[14])
Usually, when a double knot opens, the distance between the two parts slowly increases. Eventually, they will separate to the point that they are not connected enough to be considered a complete knot either regarding the laws of tzitzit[15] or the laws of making or undoing knots on Shabbat.[16] It is difficult to delineate or illustrate when exactly this occurs. When in doubt, it is proper, although not immediately critical,[17] to tighten the knot during the week. However, one should be careful not to tighten the double knots on Shabbat in such a way that he might be turning a simple single knot into a halachically significant double knot.[18] [19]