My question is with regard to the meaning of, and/or significance of, each of the different styles of tying.

The symbolic meaning of the tzitzit windings with tekhelet is stated clearly in the Gemara Menachot by Rebbi: to remind us of the heavens. Similarly does Rebbi Meir explain that the symbolic meaning of tekhelet is that of God’s throne. And so ALL the various methods of tying with tekhelet merely try to imbue that singular meaning – heaven. The reason there are so many methods is not because they all have some varying intent, but rather due to the varying interpretations of the technical implementation provided in the Gemara. This is diametrically opposed to the various methods used for tying with white only.

For once tekhelet, the color of God’s throne, was lost, then new ways of symbolically imbuing the tzitzit with the reminder of the heavens needed to be introduced. Some (like the Hida) said the wraps should equal God’s name YKVK (26), others (Arizal) said they should equal Hashem Echad (39) which also equals the number of wraps when wrapping with tekhelet.

– Mois Navon.