@surprised, you've made me think even more about "twists". Are your wrapping numbers allowed to be negative? Even though it may be impossible to specify winding sense around an isolated hole, there must be configurations of several holes where the net result of a cycle does depend on the (relative) individual senses.
Sorry i can't give you a picture, but think of an east-west double bagel - two holes and a neck. Wind the string over the north-west, under the neck and onwards under and into the eastern hole, over the top of the south-eastern limb, down again to go under the south-western limb on the western bagel and back to the start.
Now i'm fairly convinced this cycle has a different handedness from the one where all the overs and unders are reversed. So the quantum numbers need to be distinguishable (even if, as might be the case here, the two configurations have the same energy).
My next problem might be dismissed on the grounds of compactness or some other "nice" topological requirement? Find a bit of the space where two bagel-like pieces come close but don't meet. (#1) Wrap the string over the left hole, over the right, wind back down and under both to complete the cycle. Give this wrapping numbers (1,1). (#2) Wrap over the left, dip under the right, complete a figure-eight path. Give this wrapping numbers (1,-1) and think of it as #1 with a 180 twist. (#3) Make as with #1 but give it a 360 twist so the wrapping numbers are again (1,1). Now #1 and #3 are obviously not equivalent in handedness or energetically, so i guess it's anathema to string theory. If i pumped up the shape enough, the bagels would separate and the twisted bits of string would ping back and wind around *something"?
Conclusion: strings aren't allowed to twist around themselves (to the extent of touching, anyway)?