If we are willing to accept closed strings as being loops of strings moving in a background space - then I do not see any deep reason why those closed strings cannot be knotted.

Is there any deep reason for ignoring knotted closed strings ?

Think of Bosonic string theory with the Polyakov action. Can we include a topological term in one dimension higher - that captures the knottedness of the bosonic, Polyakov strings ? Think of the theta term in QCD (instantons etc). Maybe such a topological term can affect the vacuum of string theory - just as the theta term and instantons can affect the vacuum of Yang-Mills.

Indeed - is there a term constructed from the string coordinates in the background geometry, X^mu, that can be used to capture the knottedness of the string ?

One thing is for sure - a knotted string will create a Seyfert surface for a world-sheet. The world-sheet will have self-intersections etc and other topological complexities. Perhaps this will affect the compactified extra dimensions. I am sure that a propagating trefoil knot, for example, will require a much more involved world-sheet topology. Beyond Riemann surfaces and into Algebraic Surfaces in higher dimensions that may possess all sorts of complicated topology.

My suspicion is that we should not ignore the possibility of knotted strings. Maybe its something that we should think about.

Best Regards

Alwi