Many Ashkenazim now wear Tzitzith with Techeleth tied in the tradition of the RaMBa"M. However, my understanding is that Ashkenazim minimally require a double-knot at the base of the tassle in order for the Tzitzith to be Kosher, regardless of how they are tied after that, while true Yemenite Tzitzith, in the tradition of the RaMBa"M, regardless of whether or not they have Techeleth, have no double-knot at the base. May an Ashkenazi wear Tzitzith tied in the Yemenite tradition?
Rav Dov Lior rules to put Techeleth, he also rules to make double knot in the beginning and then to make ties by Rambam. I've heard this on his lesson on Mishna Berura where he showed his tzitzis tied this way.
From this I conclude that double knot is essential.
The double knot at the top is midoraisa. That's what fulfills the mitzvah The other knots developed and became custom. Some are based on the ari z"l and other kabbalistic sources but it doesn't matter if you have that first double knot. A bigger question would be wearing wool or cotton.