Applying of knot theory to string theory

[shreyakmath]

Probably a bit abstract,but I was thinking if 4D closed strings could form knots? I mean if a closed string in 4-dimensional spacetime can be considered an unknot and a knot polynomial be associated with every closed string. I also wondered that if the fundamental strings vibrate in the knotted state, their harmonics could represent something entirely different!
What are your views?

 

[ImaLooser]

Probably a bit abstract,but I was thinking if 4D closed strings could form knots? I mean if a closed string in 4-dimensional spacetime can be considered an unknot and a knot polynomial be associated with every closed string. I also wondered that if the fundamental strings vibrate in the knotted state, their harmonics could represent something entirely different!
What are your views?

Knots are always from "materials" that have one less dimension than the space in which they are embedded. So a 4-D space would have 3-D knots. I can't say that I can picture this.

 

[tom.stoer]

A knot is nothing else but the map of a closed one-dim. loop to a target space. The fact that the knot is knotted is not a property of the loop but a property of the map. It is encoded in the topological structure of the "target space minus the knot".

Now you can try to do this in N > 3 dimensions.

It turns out that there are no knots in N > 3, that means that every "knot" can be unknotted, the "target space minus the knot" is topologically identical to the "target space minus a trivial circle".

 

[someGorilla]

Strings cannot form knots in 4 dimensions. (or only trivial knots)

But this got me thinking that 2d surfaces can form "knots" in 4 dimensions. Are these, too, called knots in topology? By tom.stoer's definition for example the Klein bottle would be a knot in 4d, though not in any superior dimension.

 

[tom.stoer]

Hm, knots are always defined as 1-dim loops in 3-space. But you are right, there is this analogy in higher dim. spaces

 

[atyy]

But this got me thinking that 2d surfaces can form "knots" in 4 dimensions. Are these, too, called knots in topology? By tom.stoer's definition for example the Klein bottle would be a knot in 4d, though not in any superior dimension.

Hm, knots are always defined as 1-dim loops in 3-space. But you are right, there is this analogy in higher dim. spaces

torsten answered this in https://www.physicsforums.com/showthread.php?t=678799

 

[LotusPond_14]

what if the "string vibration" is actually a spin projection of a one dimensional string on a two dimensional plane. we cannot see the actual spin of the string but we can see the vibration behaviour on a plane transversal to the observer.

then how then can a closed loop can be formed in two dimensional space. visualise a telephone cord when you bend it and began twisting on the middle. notice that the two halves of the cord will twist in unison. then appying more spinning motion will result the beginning and end parts of the cord will meet; thus forming an irregular loop depending on how much spin is applied to the middle of the cord.

somehow, the telephone cord represent the actual string in space, has one dimension (length) but suspended in space in a spiral pattern. when a disturbance is acted upon the spiral string, it either contracts(gravity), elongates (light, sound, magnetism) or forms a dimensional loop (matter).

correct me if I am wrong but this only an observation.