It’s known as the 7-8-11-13 method. I heard rumblings of it being mentioned by the Tanchuma. Do any modern halachic authorities consider this a valid attempt at fulfilling techeiles. Also should there be a white string before every chulyo(set of loops) on this set. Or one at the beginning and one at the end? Saw something regarding that as well. All knowledge appreciated. Cheers!
The tying method of 7,8,11,13 was used after the techeiles went out if disuse(forgotten). From the gemara in Menachos 39a the numbers of windings per chulya(braid) is 3 or 7(Rabbeinu tam),and a minimum of 7 braids and maximum of 13. It would seem that method for techeiles doesn't align with gemara so well. It should be noted almost every way of tying will basically fulfill the mitzvah bdeieved. Having 2 knots and one braid of at least 3 wrappings is the minimum.
With regards to the white wimdings ,techiles windings should start with white and end with white as per the gemara in Menachos 39a
תנא כשהוא מתחיל מתחיל בלבן הכנף מין כנף וכשהוא מסיים מסיים בלבן מעלין בקודש ולא מורידין
It was taught: When one begins to form the windings, he begins winding with a white string. This is because the verse indicates that one first inserts “the fringe of the corner” (Numbers 15:38), i.e., the white strings, which are of the same type as the corner of the garment. And when he concludes the windings, he concludes with a white string, in accordance with the principle: One elevates to a higher level in matters of sanctity and does not downgrade.
I don't have the exact sources in front of me, but some points on tying:
-Tie the corners in the order front right, front left, back right, back left as there is greater importance given to the right vs left and to the front vs back. (Based on this order, it would seem that the front vs back is a bigger consideration than right vs left.)
-Say לשם מצות ציצית while holding the four strings, right before inserting them into the hole.
-Insert all four strings at the same time into the hole. (This is from the Kaf HaChaim, but some say it is only a hiddur to do so.)
-When tying Techeiles, the first wrap closest to the beged and the final wrap furthest from the beged should be white. The rest can be either white or blue, but you should follow an established method of tying. The websites linked to above both have information on the different methods of tying.
I have been tying tzitzis now for over 10 years for numerous individuals, giving me experience with all the various methods of tying for both Techeiles and plain white. I'm always happy to share the tips and tricks I've learned over the years.