One of the 39 prohibited Melachot (actions) on Shabbat is tying and untying knots. The Shulhan Aruch in Siman 317 establishes three levels of severity for this prohibition, based on the Rambam. If the knot is both permanent and professional, it is a Torah Prohibition. If it has only one of those attributes, permanent or professional, it is becomes a Rabbinic prohibition. If it is neither permanent nor professional, it is permitted.
Based on this, the Beur Halacha (Rav Yisrael Meir Kagan of Radin, 1839-1933) asks why tying Sisit is universally considered a prohibited act. The Sisit knots are not special knots, only done by professional craftsmen, and his intention can be to keep it tied until after Shabbat, and therefore it is not permanent.
He answers that although he may intend to keep it tied temporarily, most people tie Sisit knots with the intent that it stays permanently. The definition of permanent and temporary are determined by the general population, and not by the individual.
Hacham Ovadia points out that sometimes the outer knot of the double knot of the Sisit becomes loose on Shabbat. It is prohibited to tighten it. Doing so may even constitute an Issur Doraita (a Torah Prohibition), according to Rashi who does not take the level of craftsmanship required into account. Therefore, one should avoid his natural urge to tighten a loose Sisit knot on Shabbat.
SUMMARY
It is prohibited to tie the knots of Sisit on Shabbat.
