The website Sherlock visits, beecircuit.com, leads to the CBS programming website. (This is still true as of November 2022.)
Captain Gregson tells Joan the officer who would most likely take over the 11th Precinct in his absence works at the 51st Precinct in the Bronx. This precinct is fictional as the only precincts with numbers in the 50's are the 50th Precinct and the 52nd Precinct, both in the Bronx.
The hotel suite that Sheik Nassar stays was is an actual guest suite at the Waldorf Astoria Towers in NYC.
Twice Sherlock says the case is a Gordian Knot, and then says they've been given a sword with which to cut it: According to Greek mythology, when the peasant Gordius became king of Gordium in Phrygia, he dedicated his wagon to Jupiter and fastened its yoke to a beam with a very complex knot. Centuries later, when Alexander the Great arrived on the scene, he was told that he couldn't conquer and rule Asia unless he proved himself worthy by untying the knot. Alexander quickly solved his problem - and gained a new kingdom - by slicing the knot in half with his sword. Since then, "Gordian knot" has become a term for a difficult problem, and the phrase "cut the Gordian knot" has become a popular way to describe a neat solution for an apparently insurmountable difficulty