Shalom, I am Ashkenazi, so I would like to know: What method should I use to tie my Techelet? And why?
According to most of the great Halachic authorities Techelet (Biblical blue dye for Tzitzit) should not be used nowadays, because we have no tradition on this issue, and it may mislead and cause error to future generations who will think that there always was a tradition on this, and for other reasons as well. [ערוה"ש סימן ט סעיף יב, קובץ תשובות להגרי"ש אלישיב סימן ב, תשובות והנהגות ח"א סימן כט וח"ד סימן ה, אאמו"ר שליט"א בשם מורו ורבו הגרח"פ שיינברג שליט"א שאין להשתמש בתכלת משום שזה עלול לגרום לטעות לדורות הבאים כאילו יש בזה מסורת ויביא לידי מכשול, וע"ע מראי מקומות נוספים בפסקי תשובות סימן ח הערה 2]. There is a disagreement among the Rishonim whether at the time when Techelet was used, one had to put two white strings and two Techelet strings, or one Techelet and three white (ערוה"ש סימן ט סעיף ב). The Mishnah Berurah (9, 7) quotes the Taz that in the times when there was Techelet one had to put two white strings and two Techelet strings. And so does the Aruch Hashulchan (שם) decide L’halachah. But I think that those who have the custom to use Techelet also nowadays according to their Rabbis such as the Radzin Chasidim, only put one string of Techelet which is the long string that goes around the others. The method of tying Tzitzit according to the Ashkenazi custom: you thread four strings through the hole in the corner of the Tzitzit garment, while one of the four strings is longer then the rest. Then we have eight strings, four on each side. Take the four strings from each side and tie them over each other twice (a double knot). Turn the long string around the others seven times and tie again twice. Turn the long string around the others eight times and tie again twice. Turn the long string around the others eleven times and tie again twice. Turn the long string around the others thirteen times and tie again twice. In summary we have five double knots with the Tzitzit wrapped around between them according to the following numbers 7 8 11 13. The same should be done to all four corners. The meaning of the numbers is as follows: 7 & 8 stand for the sacred Name of Yud & Hey. Then 11 which combined with the above stands for the sacred Name of Havayah. Afterwards 13, which is the Gematria (assignment of numeric values to Hebrew letters) of the word Echad (one). All these together stand for the words “Hashem is one”. [שו"ע או"ח סימן יא סעיף יד, ומ"ב ס"ק ע].