Can Knotted Closed Strings Exist in Higher Dimensions?

[Alwi]

Hi all ! I am curious about the possibility of knotted strings. If we can accept the propagation of closed loop strings, then shouldn't we consider the possibility of knotted, closed loop strings ?

I do not see - offhand - any reason why a closed string cannot be knotted. Think of a propagating trefoil knot for example.

Of course - I would expect the mathematics to be much more involved. For a knotted closed string, the world-sheet is much more than a mere Riemann surface. It is now a general Algebraic Surface that may possesses self-intersections and other topological phenomena.

In this case - I would expect the knottedness of the string to have implications on the compact extra-dimensions.

I would like to hear some comments and ideas regarding this matter.

Best Regards
Alwi

 

[xepma]

The knotting of a string is topologically non-trivial only in 3 dimensions. For string theory you need a whole lot more dimensions, and so every string would be topologically equivalent to the unkot.

On the other hand, you could fool around a bit with compactified dimensions which leads to winding string around dimensions and so.

Furthermore, the concept of knotting quickly leads to concepts such as category theory and topological quantum field theory in particular. These are abstract mathematical structures and serve as a framework for string theory. So to consider "knotted strings" is in fact equivalent to studying these framework in relation to string theory.

 

[Alwi]

Dear xepma,

No I disagree with your statement that every string will be topologically equivalent to the unknot. Knottedness can take place totally in the large 3+1 dimensions. The compact extra dimensions need not unravel that knottedness. It all depends on how we do our compactification and the kinds of homotopy we want to consider. You can easily construct a knot in S^3 x G where G is a compact space. Just do your knotting in S^3 and trivially multiply by G. Can you find a homotopy in S^3 x G that unravels this knot ? Yes and no. If you restrict yourself to homotopies entirely done in S^3 - then of course you cannot unravel the knot.

I think the issue that I want to raise here - to be very precise - is whether we should :

1) Include a knot topological term in the Polyakov action (for Bosonic strings) and

2) Include self-intersecting Algebraic Surfaces of knot world-sheets -- when performing a sum-over-surfaces a-la Polyakov's path integral approach.

Best Regards
Alwi

 

[Alwi]

Dear Xepma,

Here is also another reason why I think knotted strings ought to be considered. Its not clear to me that one can always find a homotopy involving the compact extra dimension - that will unknot the knotted string. What makes you think that there will be no obstructions ? Maybe the compact extra dimensions possesses holes (as required for the number of fermion generations) etc which makes some types of homotopy impossible.

So you see - knotted strings are not necessarily trivial by virtue of the extra dimensions.

Best Regards
Alwi