Since I'm a Divinity School student, there will be citations with links to the texts.
So in the Torah it says
7 Remember the sabbath day, to keep it holy. 8 Six days shalt thou labour, and do all thy work; 9 but the seventh day is a sabbath unto the LORD thy God, in it thou shalt not do any manner of work, thou, nor thy son, nor thy daughter, nor thy man-servant, nor thy maid-servant, nor thy cattle, nor thy stranger that is within thy gates; 10 for in six days the LORD made heaven and earth, the sea, and all that in them is, and rested on the seventh day; wherefore the LORD blessed the sabbath day, and hallowed it. (Exodus 20:7-10).
There's also the other set of 10 commandments in Deuteronomy, which has the similar
11 Observe the sabbath day, to keep it holy, as the LORD thy God commanded thee. 12 Six days shalt thou labour, and do all thy work; 13 but the seventh day is a sabbath unto the LORD thy God, in it thou shalt not do any manner of work, thou, nor thy son, nor thy daughter, nor thy man-servant, nor thy maid-servant, nor thine ox, nor thine ass, nor any of thy cattle, nor thy stranger that is within thy gates; that thy man-servant and thy maid-servant may rest as well as thou. 14 And thou shalt remember that thou was a servant in the land of Egypt, and the LORD thy God brought thee out thence by a mighty hand and by an outstretched arm; therefore the LORD thy God commanded thee to keep the sabbath day.(Deut. 5:11-14)
So this is a bit ambiguous in that "any manner of work" is left undefined. Luckily, the Rabbis, writing several centuries later, define "work" as creative work, since God created the universe in 6 days and rested on the 7th, therefore in remembrance of that we should stop creating. There's a specific set of things that are prohibited, 39 of them, to be exact. Here is a list of them, with some explanation. Note how knitting is there.
Some things are debateable, such as electricity (which later rabbis prohibited the use of on Shabbat), or carrying an umbrella,1 but knitting is really creative work. One might argue that knitting is relaxing and Shabbat is supposed to be a day of rest, and I would agree with that, except that I'm creating something when I knit and I'm not supposed to be creating anything.
And that is why Sabbath-observant Jews do not knit on the Sabbath.
Though I did find out that in Israel they held International Knit In Public Day on Sunday the 15th of June, though that wouldn't have helped me since I was not in Israel, and even if I did knit in public that day I was alone in my knittingness.
1This one is a little abstract- once upon a time you used to have to put together umbrellas, much like the ones that now are on patio tables. So the assembling of it was categorized under "erecting a shelter" and carrying umbrellas was prohibited, even though you could have made it before Shabbat, since people might see it and think it was okay to put together an umbrella on Shabbat. Then they invented modern umbrellas, which don't need to be assembled. But the rabbis thought that people still might look at it and dthink you could assemble an umbrella on Shabbat. And some said that opening it was also a problem. Even if you walked around with an umbrella which had a sign on it which said "this umbrella was opened before Shabbat" the rabbis still prohibit using an umbrella. At this point I think it's just a matter of tradition and not wanting to change it, but it is a very silly law to still be following.