As for your son’s concern about tying knots on the Sabbath, it is commendable that he is being conscientious about observing the laws of Shabbat. The prohibition against tying knots on the Sabbath is based on the 39 melachot, or labors, used for the construction of the Tabernacle, one of which includes tying and untying knots. Different authorities have varying opinions on what constitutes a permanent knot, with some considering tightly squeezed knots as permanent as well.
It is important to note that while there may not be a sin offering for violating these laws today, there are still consequences, such as invalidating one’s testimony as a halachic witness, which can have implications in matters such as betrothals or kiddushin.
In terms of your concern about your son’s appearance on the Sabbath, it is understandable that you want him to present himself well, especially in the context of potential shidduchim. However, it is also important to prioritize his observance of halacha and respect for the laws of Shabbat. It might be worth discussing with him how he can maintain a neat appearance while still adhering to the laws regarding tying knots on the Sabbath.
Ultimately, it is commendable that your son is taking his observance of Shabbat seriously, and it is important to support him in his commitment to observing halacha while also addressing any concerns about his appearance in a thoughtful and respectful manner.
However, if his intent was to untie on the very same day that it was tied, then no violation is encountered, neither biblical nor rabbinical, in his untying it. We see this view clearly in the text of Tur (ad loc.).
(To be continued)