The 39 Melachot

Following the detailed description of the work (melacha) entailed in the building and running of the Mishkan (Tabernacle), Hashem commands Moshe regarding the observance of Shabbat:

Speak also to the Children of Israel saying, “Verily my Sabbaths shall you keep; for it is a sign between me and you for all generations that you may know that I am the Lord that sanctifies you…for whoever does any work (melacha) on it, that soul shall be cut off from among his people…(Exodus 31:13-14)

The juxtaposition of the two topics, and the common usage of the word melacha between them, is the basis for deriving which categories of work are prohibited on Shabbat. In essence, all the work which was necessary for building/running the Mishkan defines that which we must refrain from doing on Shabbat. The Talmud describes 39 such work categories, several of which pertain to tekhelet:

Shearing wool.
Laundering the wool.
Carding the wool.
Dyeing the wool.
Spinning the wool into thread.
Arranging the threads on a loom.
Making rings/loops for the loom.
Weaving the thread into cloth.
Unraveling/breaking threads.
Tying a knot.
Untying a knot.

According to the Talmud, tying and untying knots specifically refers to work performed while obtaining the chilazon for producing the tekhelet dye. Fishermen would sometimes have to take rope from one net, and add it to another net, by way of untying and tying knots.