Sometimes when I check my Tzitzis on Shabbos I find that one of the knots was loose and it has become undone. What should I do in such a case? Can I still wear the Talis / Talis Katon? Can I redo the knot very lightly in a way that it will not last, like just loop the two sides into each other without tightening?

Leave it. The Tzitzis may be worn as is. Do not redo it even lightly.   Sources: אסור להדק הקשר וי״א שחייב חטאת - קצוה״ש קכג בבדה״ש י. וראה דע״ת…

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