Among the thirty-nine Melachot (categories of activity forbidden on Shabbat) are "Kosheir" and "Matir" tying and untying knots. According to the practice of the Sepharadim, based on the ruling of the Shulchan Aruch, one may not tie on Shabbat any knot that he intends on leaving tied for more than seven days (Shulchan Aruch siman 317:1). Some people are accustomed to leaving their neckties permanently tied and then just slipping them on and adjusting them. A person may not tie a necktie on Shabbat with the intention of leaving it permanently tied, despite the fact that a necktie is not a professional knot. By the same token, it is forbidden to untie a necktie that had been tied with the intention that it should remain tied for more than seven days. If a necktie had been tied during the week with this intent, one may place it around his neck on Shabbat and adjust it, since this does not constitute tying.
Is it permissible to tie a double knot on Shabbat?
Although Ashkenazim follow the stringent view on this issue, the Hid"a (Rabbi Chayim Yosef David Azulai, Israel, 1724-1806), in siman 317 Ot 1, ruled that one may, in fact, tie a double knot on Shabbat, provided, of course, that he does not intend to leave the knot tied for more than seven days. Therefore, it is permissible for Sepharadim to tie a child's shoelaces in a double knot on Shabbat, and, by the same token, to untie a double knot on Shabbat. (See Menuhat Ahava, Helek 2, page 27-29.)
Is it permissible on Shabbat to tie or untie "twisties" used for closing sandwich bags?
Halacha does not consider wrapping a twisty "tying" with respect to the prohibitions of Kosheir and Matir, and therefore one may tie or untie twisties on Shabbat with intent to untie it within 7 days. (Menuhat Ahava, ibid, page 38-39.)
Summary: One may not tie even a simple knot on Shabbat such as tying a necktie if he intends to leave it tied for more than seven days, nor may one untie a knot that had been tied with this intent. One may adjust a necktie on Shabbat, even if it had been tied with the intention that it would be left tied permanently. Ashkenazim forbid tying or untying a double knot on Shabbat, whereas Sepharadim are lenient in this regard. One may tie or untie "twisties" on Shabbat.